## 1. The Aims of Statistical Mechanics (SM)



Statistical Mechanics (SM) is the third pillar of modern physics, next
to quantum theory and relativity theory. Its aim is to account for the
macroscopic behaviour of physical systems in terms of dynamical laws
governing the microscopic constituents of these systems and the
probabilistic assumptions made about them. One aspect of that
behaviour is the focal point of SM: equilibrium. Much of SM
investigates questions concerning equilibrium, and philosophical
discussions about SM focus on the foundational assumptions that are
employed in answers to these questions.



Let us illustrate the core questions concerning equilibrium with a
standard example. Consider a gas confined to the left half of a
container with a dividing wall (see
 Figure 1a).
 The gas is in *equilibrium* and there is no manifest change in
any of its macro properties like pressure, temperature, and volume.
Now you suddenly remove the dividing wall (see
 Figure 1b),
 and, as result, the gas starts spreading through the entire available
volume. The gas is now no longer in equilibrium (see
 Figure 1c).
 The spreading of the gas comes to an end when the entire available
space is filled evenly (see
 Figure 1d).
 At this point, the gas has reached a new equilibrium. Since the
process of spreading culminates in a new equilibrium, this process is
an *approach to equilibrium*. A key characteristic of the
approach to equilibrium is that it seems to be *irreversible*:
systems move from non-equilibrium to equilibrium, but not *vice
versa*; gases spread to fill the container evenly, but they do not
spontaneously concentrate in the left half of the container. Since an
irreversible approach to equilibrium is often associated with
thermodynamics, this is referred to as *thermodynamic
behaviour*. Characterising the state of equilibrium and accounting
for why, and how, a system approaches equilibrium is the core task for
SM. Sometimes these two problems are assigned to separate theories (or
separate parts of a larger theory), which are then referred to as
*equilibrium SM* and *non-equilibrium SM*,
respectively.




![a rectangle one half blue and one half white with a solid black line between the two halves](figure1a.svg)

(a)




![same rectangle with no black line between the two halves](figure1b.svg)

(b)




![same rectangle but colors now in a gradient from blue at one end to white at the other](figure1c.svg)

(c)




![same rectangle but now a solid light blue](figure1d.svg)

(d)




Figure 1 [An
 extended description of figure 1
 is in the supplement.]




While equilibrium occupies centre stage, SM of course also deals with
other issues such as phase transitions, the entropy costs of
computation, and the process of mixing substances, and in
philosophical contexts SM has also been employed to shed light on the
nature of the direction of time, the interpretation of probabilities
in deterministic theories, the state of the universe shortly after the
big bang, and the possibility of knowledge about the past. We will
touch on all these below, but in keeping with the centrality of
equilibrium in SM, the bulk of this entry is concerned with an
analysis of the conceptual underpinnings of both equilibrium and
non-equilibrium SM.



Sometimes the aim of SM is said to provide a reduction of the laws of
thermodynamics: the laws of TD provide a correct description of the
macroscopic behaviour of systems and the aim of SM is to account for
these laws in microscopic terms. We avoid this way of framing the aims
of SM. Both the nature of reduction itself, and the question whether
SM can provide a reduction of TD (in some specifiable sense) are
matters of controversy, and we will come back to them in
 Section 7.5.


## 2. The Theoretical Landscape of SM



Philosophical discussions in SM face an immediate difficulty.
Philosophical projects in many areas of physics can take an accepted
theory and its formalism as their point of departure. Philosophical
discussions of quantum mechanics, for instance, can begin with the
Hilbert space formulation of the theory and develop their arguments
with reference to it. The situation in SM is different. Unlike
theories such as quantum mechanics, SM has not yet found a generally
accepted theoretical framework or a canonical formalism. What we
encounter in SM is a plurality of different approaches and schools of
thought, each with its own mathematical apparatus and foundational
assumptions. For this reason, a review of the philosophy of SM cannot
simply start with a statement of the theory's basic principles
and then move on to different interpretations of the theory. Our task
is to first classify different approaches and then discuss how each
works; a further question then concerns the relation between them.



Classifying and labelling approaches raises its own issues, and
different routes are possible. However, SM's theoretical
plurality notwithstanding, most of the approaches one finds in it can
be brought under one of three broad theoretical umbrellas. These are
known as "Boltzmannian SM" (BSM), the "Boltzmann
Equation" (BE), and "Gibbsian SM" (GSM). The label
"BSM" is somewhat unfortunate because it might suggest
that Boltzmann, only (or primarily) championed this particular
approach, whereas he has in fact contributed to the development of
many different theoretical positions (for an overview of his
contributions to SM see the entry on
 Boltzmann's work in statistical physics;
 for detailed discussions see Cercignani (1998), Darrigol (2018), and
Uffink (2007)). These labels have, however, become customary and so we
stick with "BSM" despite its historical infelicity. We
will now discuss the theoretical backdrop against which these
positions are formulated, namely dynamical systems, and then introduce
the positions in
 SS4,
 SS5, and
 SS6,
 respectively. Extensive synoptic discussion of SM can also be found
in Frigg (2008b), Shenker (2017a, 2017b), Sklar (1993), and Uffink
(2007).


## 3. Dynamical Systems



Before delving into the discussion of SM, some attention needs to be
paid to the "M" in SM. The mechanical background theory
against which SM is formulated can be either classical mechanics or
quantum mechanics, resulting in either classical SM or quantum SM.
Foundational debates are by and large conducted in the context of
classical SM. We follow this practice in the current entry, but we
briefly draw attention to problems and issues that occur when moving
from a classical to a quantum framework
 (SS4.8).
 From the point of view of classical mechanics, the systems of
interest in SM have the structure of *dynamical system*, a
triple \((X,\) \(\phi,\) \(\mu).\) \(X\) is the state space of the
system (and from a mathematical point of view is a set). In the case
of a gas with \(n\) molecules this space has \(6n\) dimensions: three
coordinates specifying the position and three coordinates specifying
the momentum of each molecule. \(\phi\) is the time evolution
function, which specifies how a system's state changes over
time, and we write \(\phi\_{t}(x)\) to denote the state into which
\(x\) evolves after time \(t\). If the dynamic of the system is
specified by an equation of motion like Newton's or
Hamilton's, then \(\phi\) is the solution of that equation. If
we let time evolve, \(\phi\_{t}(x)\) draws a "line" in
\(X\) that represents the time evolution of a system that was
initially in state \(x\); this "line" is called a
*trajectory*. Finally, \(\mu\) is a measure on \(X\), roughly a
means to say how large a part of \(X\) is. This is illustrated
schematically in
 Figure 2.
 For a more extensive introductory discussion of dynamical systems see
the entry on
 the ergodic hierarchy, section on dynamical systems,
 and for mathematical discussions see, for instance, Arnold and Avez
(1967 [1968]) and Katok and Hasselblatt (1995).



![A green rectangle, link to extended description below](figure2.svg)

Figure 2 [An
 extended description of figure 2
 is in the supplement.]




It is standard to assume that \(\phi\) is deterministic, meaning, that
every state \(x\) has exactly one past and exactly one future, or, in
geometrical terms, that trajectories cannot intersect (for a
discussion of determinism see Earman (1986)). The systems studied in
BSM are such that the volume of "blobs" in the state space
is conserved: if we follow the time evolution of a "blob"
in state space, this blob can change its shape but not its volume.
From a mathematical point of view, this amounts to saying that the
dynamics is *measure-preserving*: \(\mu(A) = \mu(\phi\_{t}(A))\)
for all subsets \(A\) of \(X\) and for all times \(t\). Systems in SM
are often assumed to be governed by Hamilton's equations of
motion, and it is a consequence of Liouville's theorem that the
time evolution of a Hamiltonian system is measure-preserving.


## 4. Boltzmannian Statistical Mechanics (BSM)



In the current debate, "BSM" denotes a family of positions
that take as their starting point the approach that was first
introduced by Boltzmann in his 1877 paper and then presented in a
streamlined manner by Ehrenfest and Ehrenfest-Afanassjewa in their
1911 [1959] review. In this section we discuss different contemporary
articulations of BSM along with the challenges they face.


### 4.1 The Framework of BSM



To articulate the framework of BSM, we distinguish between
micro-states and macro-states; for a discussion of the this framework
see, for instance, Albert (2000), Frigg (2008b), Goldstein (2001), and
Sklar (1993). The *micro-state* of a system at time \(t\) is
the state \(x \in X\) in which the system is at time \(t\). This state
specifies the exact mechanical state of every micro-constituent of the
system. As we have seen in the previous section, in the case of a gas
\(x\) specifies the positions and momenta of every molecule in the
gas. Intuitively, the *macro-state* \(M\) of a system at time
\(t\) specifies the macro-constitution of the system at \(t\) in
terms of variables like volume, temperature and other properties
measurable, loosely speaking, at human scales, although, as we will
see in
 Section 4.8,
 reference to thermodynamic variables in this context must be taken
with a grain of salt. The configurations shown in
 Figure 1
 are macro-states in this sense.



The core posit of BSM is that macro-states supervene on
micro-states, meaning that any change in the system's
macro-state *must* be accompanied by a change in the
system's micro-state: every micro-state \(x\) has exactly one
corresponding macro-state \(M\). This rules out that, say, the
pressure of a gas can change while the positions and momenta each of
its molecules remain the same (see entry on
 supervenience).
 Let \(M(x)\) be the unique macro-state that corresponds to microstate
\(x\). The correspondence between micro-states and macro-states
typically is not one-to-one and macro-states are multiply realisable.
If, for instance, we swap the positions and momenta of two molecules,
the gas' macro-state does not change. It is therefore natural to
group together all micro-states \(x\), that correspond to the same
macro-state \(M\):

 
\[X\_{M} = \{ x \in X \text{ such that } M(x) = M\}.\]


\(X\_{M}\) is the *macro-region* of \(M\).



Now consider a complete set of macro-states (i.e., a set that contains
every macro-state that the system can be in), and assume that there
are exactly \(m\) such states. This complete set is \(\{
M\_{1},\ldots,M\_{m}\}\). It is then the case that the corresponding set
of macro-regions, \(\{ X\_{M\_{1}},\ldots,X\_{M\_{m}}\}\), forms a
partition of \(X\), meaning that the elements of the set do not
overlap and jointly cover \(X\). This is illustrated in
 Figure 3.



![A more complex diagram containing elements of figures 1 and 2, link to extended description below](figure3.svg)

Figure 3 [An
 extended description of figure 3
 is in the supplement.]




The figure also indicates that if the system under study is a gas,
then the macro-states correspond to different states of the gas we
have seen in
 Section 1.
 Specifically, one of the macro-states corresponds to the initial
state of the gas, and another one corresponds to its final equilibrium
state.



This raises two fundamental questions that occupy centre stage in
discussions about BSM. First, what are macro-states and how is the
equilibrium state identified? That is, where do we get the set
\(\{M\_{1},\ldots,M\_{m}\}\) from and how do we single out one member of
the set as the equilibrium macro-state? Second, as already illustrated
in
 Figure 3,
 an approach to equilibrium takes place if the time evolution of the
system is such that a micro-state \(x\) in a non-equilibrium
macro-region evolves such that \(\phi\_{t}(x)\) lies in the equilibrium
macro-region at a later point in time. Ideally one would want this to
happen for all \(x\) in any non-equilibrium macro-region, because this
would mean that all non-equilibrium states would eventually approach
equilibrium. The question now is whether this is indeed the case, and,
if not, what "portion" of states evolves differently.



Before turning to these questions, let us introduce the *Boltzmann
entropy* \(S\_{B}\), which is a property of a macro-state defined
through the measure of the macro-states' macro-region:



\[S\_{B}(M\_{i}) = k\log{\lbrack\mu(X\_{M\_{i}}})\rbrack\]


for all \(i = 1,\ldots, m\), where \(k\) is the so-called Boltzmann
constant. Since the logarithm is a monotonic function, the larger the
measure \(\mu\) of a macro-region, the larger the entropy of the
corresponding macro-state.



This framework is the backbone of positions that self-identify as
"Boltzmannian". Differences appear in how the elements of
this framework are articulated and in how difficulties are
resolved.


### 4.2 Defining Equilibrium: Boltzmann's Combinatorial Argument



An influential way of defining equilibrium goes back to Boltzmann
(1877); for contemporary discussion of the argument see, for instance,
Albert (2000), Frigg (2008b), and Uffink (2007). The approach first
focusses on the state space of one particle of the system, which in
the case of a gas has six dimensions (three for the particle's
positions in each spatial dimension and a further three for the
corresponding momenta). We then introduce a grid on this
space--an operation known as *coarse-graining*--and
say that two particles have the same *coarse-grained
micro-state* if they are in the same grid cell. The state of the
entire gas is then represented by an *arrangement*, a
specification of \(n\) points on this space (one for each particle in
the gas). But for the gas' macro-properties it is irrelevant
which particle is in which state, meaning that the gas'
macro-state must be unaffected by a permutation of the particles. All
that the macro-state depends on is the *distribution* of
particles, a specification of how many particles are in each grid
cell.



The core idea of the approach is to determine how many arrangements
are compatible with a given distribution, and to define the
equilibrium state as the one for which this number is maximal. Making
the strong (and unrealistic) assumption that the particles in the gas
are non-interacting (which also means that they never collide) and
that the energy of the gas is preserved, Boltzmann offered a solution
to this problem and showed that the distribution for which the number
of arrangements is maximal is the so-called discrete Maxwell-Boltzmann
distribution:

 
\[n\_{i} = \alpha\exp\left({-\beta} E\_{i} \right),\]


where \(n\_{i}\) is the number of particles in cell \(i\) if the
coarse-graining, \(E\_{i}\) is the energy of a particle in that cell,
and \(\alpha\) and \(\beta\) are constants that depend on the number
of particles and the temperature of the system (Tolman 1938 [1979]:
Ch. 4). From a mathematical point of view, deriving this distribution
is a problem in combinatorics, which is why the approach is now known
as the *combinatorial argument*.



As Paul and Tatiana Ehrenfest pointed out in their 1911 [1959] review,
the mathematical structure of the argument also shows that if we now
return to the state space \(X\) of the entire system (which, recall,
has \(6n\) dimensions), the macro-region of the equilibrium state thus
defined is the largest of all macro-regions. Hence, the equilibrium
macro-state is the macro-state with the largest macro-region. In
contemporary discussions this is customarily glossed as the
equilibrium macro-state not only being larger than any other
macro-state, but as being enormously larger and in fact taking up most
of \(X\) (see, for instance, Goldstein 2001). However, as Lavis (2008)
points out, the formalism only shows that the equilibrium macro-region
is larger than any other macro-region and it is not a general truism
that it takes up most of the state space; there are in fact systems in
which the non-equilibrium macro-regions taken together are larger than
the equilibrium macro-region.



Since, as we have seen, the Boltzmann entropy is a monotonic function
of the measure of a macro-region, this implies that the equilibrium
microstate is also the macro-state with the largest Boltzmann entropy,
and the approach to equilibrium is a process that can be characterised
by an increase of entropy.



Two questions arise: first, is this a tenable general definition of
equilibrium, and, second, how does it explain the approach to
equilibrium? As regards the first question, Uffink (2007) highlights
that the combinatorial argument assumes particles to be
non-interacting. The result can therefore be seen as a good
approximation for dilute gases, but it fails to describe (even
approximately) interacting systems like liquids and solids. But
important applications of SM are to systems that are not dilute gases
and so this is a significant limitation. Furthermore, from a
conceptual point of view, the problem is that a definition of
equilibrium in terms of the number of arrangements compatible with a
distribution makes no contact with the thermodynamic notion of
equilibrium, where equilibrium is defined as the state to which an
isolated system converges when left to itself (Werndl & Frigg
2015b). Finally, this definition of equilibrium is completely
disconnected form the system's dynamics, which has the odd
consequence that it would still provide an equilibrium state even if
the system's time evolution was the identity function (and hence
nothing ever changed and no approach to equilibrium took place). And
even if one were to set thermodynamics aside, there is nothing truly
macro about the definition, which in fact directly constructs a
macro-region without ever specifying a macro-state.



A further problem (still as regards the first question) is the
justification of coarse-graining. The combinatorial argument does not
get off the ground without coarse-grained micro-states, and so the
question is what legitimises the use of such states. The problem is
accentuated by the facts that the procedure only works for particular
kind of coarse-graining (namely if the grid is parallel to the
position and momentum axes) and that the grid cannot be eliminated by
taking a limit which lets the grid size tend toward zero. A number of
justificatory strategies have been proposed but none is entirely
satisfactory. A similar problem arises with coarse-gaining in Gibbsian
SM, and we refer the reader to
 Section 6.5
 for a discussion.



As regards the second question, the combinatorial argument itself is
silent about why and how systems approach equilibrium and additional
ingredients must be added to the account to provide such an
explanation. Before discussing some of these ingredients (which is the
topic of much of the remainder of this section), let us discuss two
challenges that every explanation of the approach to equilibrium must
address: the reversibility problem and the recurrence problem.


### 4.3 Two Challenges: Reversibility and Recurrence



In
 Section 2
 we have seen that at bottom the physical systems of BSM have the
structure of a dynamical system \((X,\) \(\phi,\) \(\mu)\) where
\(\phi\) is deterministic and measure preserving. Systems of this kind
have two features that pose a challenge for an understanding of the
approach to equilibrium.



The first feature is what is known as *time-reversal
invariance*. Intuitively you can think of the time-reversal of a
process as what you get when you play a movie of a process backwards.
The dynamics of system is time-reversal invariant if every process
that is allowed to happen in one direction of time is also allowed to
happen the reverse direction of time. That is, for every process that
is allowed by the theory it is that case that if you capture the
process in a movie, then the process that you see when you play the
movie backwards is also allowed by the theory; for detailed and more
technical discussions see, for instance, Earman (2002), Malament
(2004), Roberts (2022), and Uffink (2001).



Hamiltonian systems are time-reversal invariant and so the most common
systems studied in SM have this property. A look at
 Figure 3
 makes the consequences of this for an understanding of the approach
to equilibrium clear. We consider a system whose micro-state initially
lies in a non-equilibrium macro-region and then evolves into a
micro-state that lies in the equilibrium macro-region. Obviously, this
process ought to be allowed by the theory. But this means that the
reverse process--a process that starts in the equilibrium
macro-region and moves back into the initial non-equilibrium macro
region--must be allowed too. In
 Section 1
 we have seen that the approach to equilibrium is expected to be
*irreversible*, prohibiting systems like gases to spontaneously
leave equilibrium and evolve into a non-equilibrium state. But we are
now faced with a contradiction: if the dynamics of the system is
time-reversal invariant, then the approach to equilibrium cannot be
irreversible because the evolution from the equilibrium state to a
non-equilibrium state is allowed. This observation is known as
*Loschmidt's reversibility objection* because it was first put
forward by Loschmidt (1876); for a historical discussion of this
objection, see Darrigol (2021).



The second feature that poses a challenge is *Poincare
recurrence*. The systems of interest in BSM are both
measure-preserving and spatially bounded: they are gases in a box,
liquids in a container and crystals on a laboratory table. This means
that the system's micro-state can only access a finite region in
\(X\). Poincare showed that dynamical systems of this kind
must, at some point, return arbitrarily close to their initial state,
and, indeed do so infinitely many times. The time that it takes the
system to return close to its initial condition is called the
*recurrence time*. Like time-reversal invariance,
*Poincare recurrence* contradicts the supposed
irreversibility of the approach to equilibrium: it implies that
systems will return to non-equilibrium states at some point. One just
has to wait for long enough. This is known as *Zermelo's
recurrence objection* because it was first put forward by Zermelo
(1896); for a historical discussion see Uffink (2007).



Any explanation of the approach to equilibrium has to address these
two objections.


### 4.4 The Ergodic Approach



A classical explanation of the approach to equilibrium is given within
ergodic theory. A system is ergodic iff, in the long run (i.e., in the
limit of time \(t \rightarrow \infty\)), for almost all initial
conditions it is the case that the fraction of time that the
system's trajectory spends in a region \(R\) of \(X\) is equal
to the fraction that \(R\) occupies in \(X\) (Arnold & Avez 1967
[1968]). For instance, if \(\mu(R)/\mu(X) = 1/3,\) then an ergodic
system will, in the long run, spend 1/3 of its time in \(R\) (for a
more extensive discussion of ergodicity see entry on
 the ergodic hierarchy).



In
 Section 4.2
 we have seen that if the equilibrium macro-region is constructed with
the combinatorial argument, then it occupies the largest portion of
\(X\). If we now also assume that the system is ergodic, it follows
immediately that the system spends the largest portion of time in
equilibrium. This is then often given a probabilistic gloss by
associating the time that a system spends in a certain part of \(X\)
with the probability of finding the system in that part of \(X\), and
so we get that we are overwhelmingly likely to find that system in
equilibrium; for a discussion of this approach to probabilities see
Frigg (2010) and references therein.



The ergodic approach faces a number of problems. First, being ergodic
is a stringent condition that many systems fail to meet. This is a
problem because among those systems are many to which SM is
successfully applied. For instance, in a solid the molecules oscillate
around fixed positions in a lattice, and as a result the phase point
of the system can only access a small part of the energy hypersurface
(Uffink 2007: 1017). The Kac Ring model and a system of anharmonic
oscillators behave thermodynamically but fail to be ergodic (Bricmont
2001). And even the ideal gas--supposedly the paradigm system of
SM--is not ergodic (Uffink 1996b: 381). But if core-systems of SM
are not ergodic, then ergodicity cannot provide an explanation for the
approach to equilibrium, at least not one that is applicable across
the board (Earman & Redei 1996; van Lith 2001). Attempts
have been made to improve the situation through the notion of
epsilon-ergodicity, where a system is epsilon-ergodic if it is ergodic
only on subset \(Y \subset X\) where \(\mu(Y) \geq 1 - \varepsilon\),
for small positive real number \(\varepsilon\) (Vranas 1998). While
this approach deals successfully with some systems (Frigg & Werndl
2011), it is still not universally applicable and hence remains silent
about large classes of SM systems.



The ergodic approach accommodates Loschmidt's and
Zermelo's objections by rejecting the requirement of strict
irreversibility. The approach insists that systems, can, and actually
do, move away from equilibrium. What SM should explain is not strict
irreversibility, but the fact that systems spend most of the time in
equilibrium. The ergodic approach does this by construction, and only
allows for brief and infrequent episodes of non-thermodynamic
behaviour (when the system moves out of equilibrium). This response is
in line with Callender (2001) who argues that we should not take
thermodynamics "too seriously" and see its strictly
irreversible approach to equilibrium as an idealisation that is not
empirically accurate because physical systems turn out to exhibit
equilibrium fluctuations.



A more technical worry is what is known as the *measure zero
problem*. As we have seen, ergodicity says that "almost all
initial conditions" are such that the fraction of time spent in
\(R\) is equal to the fraction \(R\) occupies in \(X\). In technical
terms this means that set of initial conditions for which this is not
the case has measure zero (with respect to \(\mu\)). Intuitively this
would seem to suggest that these conditions are negligible. However,
as Sklar (1993: 182-88) points out, sets of measure zero can be
rather large (remember that set of rational numbers has measure zero
in the real numbers), and the problem is to justify why a set of
measure zero really is negligible.


### 4.5 Typicality



An alternative account explains the approach to equilibrium in terms
of *typicality*. Intuitively something is typical if it happens
in the "vast majority" of cases: typical lottery tickets
are blanks, and in a typical series of a thousand coin tosses the
ratio of the number of heads and the number of tails is approximately
one. The leading idea of a typicality-based account of SM is to show
that thermodynamic behaviour is typical and is therefore to be
expected. The typicality account comes in different version, which
disagree on how exactly typicality reasoning is put to use; different
versions have been formulated, among others, by Goldstein (2001),
Goldstein and Lebowitz (2004), Goldstein, Lebowitz, Tumulka, and
Zanghi (2006), Lebowitz (1993a, 1993b), and Volchan (2007). In
its paradigmatic version, the account builds on the observation
(discussed in
 Section 4.2)
 that the equilibrium macro-region is so large that \(X\) consists
almost entirely of equilibrium micro-states, which means that
equilibrium micro-states are typical in \(X\). The account submits
that, for this reason, a system that starts its time-evolution in a
non-equilibrium state can simply not avoid evolving into a typical
state--i.e., an equilibrium state--and staying there for
very long time, which explains the approach to equilibrium.



Frigg (2009, 2011) and Uffink (2007) argue that from the point of view
of dynamical systems theory this is unjustified because there is no
reason to assume that micro-states in an atypical set have to evolve
into a typical set without there being any further dynamical
assumptions in place. To get around this problem Frigg and Werndl
(2012) formulate a version of the account that takes the dynamics of
the system into account. Lazarovici and Reichert (2015) disagree that
such additions are necessary. For further discussions of the use of
typicality in SM, see Badino (2020), Bricmont (2022), Chibbaro,
Rondoni and Vulpiani (2022), Crane and Wilhelm (2020), Goldstein
(2012), Hemmo and Shenker (2015), Luczak (2016), Maudlin (2020),
Reichert (forthcoming), and Wilhelm (2022). As far as
Loschmidt's and Zermelo's objections are concerned, the
typicality approach has to make the same move as the ergodic approach
and reject strict irreversibility as a requirement.


### 4.6 The Mentaculus and the Past-Hypothesis



An altogether different approach has been formulated by Albert (2000).
This approach focusses on the internal structure of macro-regions and
aims to explain the approach to equilibrium by showing that the
probability for system in a non-equilibrium macro-state to evolve
toward a macro-state of higher Boltzmann entropy is high. The basis
for this discussion is the so-called *statistical postulate*.
Consider a particular macro-state \(M\) with macro-region \(X\_{M}\)
and assume that the system is in macro-state \(M\). The postulate then
says that for any subset \(A\) of \(X\_{M}\) the probability of finding
the system's micro-state in \(A\) is \({\mu(A)/\mu(X}\_{M})\). We
can now separate the micro-states in \(X\_{M}\) into those that evolve
into a higher entropy macro-state and those that move toward
macro-states of lower entropy. Let's call these sets
\(X\_{M}^{+}\) and \(X\_{M}^{-}\). The statistical postulate then says
that the probability of a system in \(M\) evolving toward a higher
entropy macro-state is \({\mu(X}\_{M}^{+})/\mu(X\_{M})\).



For it to be likely that system approaches equilibrium this
probability would have to be high. It now turns out that for purely
mathematical reasons, if the system is highly likely to evolve toward
a macro-state of higher entropy, then it is also highly likely to have
evolved into the current macro-state \(M\) from a macro-state of high
entropy. In other words, if the entropy is highly likely to increase
in the future, it is also highly likely to have decreased in the past.
Albert suggests solving this problem by regarding the entire universe
as the system being studied and then conditionalizing on the
*Past-Hypothesis*, which is the assumption that



> 
> 
> that the world first came into being in whatever particular
> low-entropy highly condensed big-bang sort of macrocondition it is
> that the normal inferential procedures of cosmology will eventually
> present to us. (2000: 96)
> 
> 
> 



Let \(M\_{p}\) be the *past state*, the state in which the world
first came into being according to the Past-Hypothesis, and let
\(I\_{t} = \phi\_{t}(X\_{M\_{p}}) \cap X\_{M}\) be the intersection of the
time-evolved macro-region of the past state and the current
macro-state. The probability of high entropy future is then
\({\mu(I\_{t} \cap X}\_{M}^{+})/\mu(I\_{t})\). If we further assume
"abnormal" states with low entropy futures are scattered
all over \(X\_{M}\), then a high entropy future can be highly likely
without it a high entropy past also being highly likely.



This approach to SM is based on three core elements: the deterministic
time evolution of the system given by \(\phi\_{t}\), the
Past-Hypothesis, and the statistical postulate. Together they result
in the assignment of a probability to propositions about the history
of a system. Albert (2015) calls this assignment the
*Mentaculus*. Albert regards the Mentaculus not only as an
account of thermodynamic phenomena, but as the backbone of a complete
scientific theory of the universe because the Mentaculus assigns
probabilities to propositions in all sciences. This raises all kind of
issues about the nature of laws, reduction, and the status of the
special sciences, which are discussed, for instance, in Frisch (2011),
Hemmo and Shenker (2021) and Myrvold and others (2016).



Like the ergodic approach, the Mentaculus must accommodate
Loschmidt's and Zermelo's objections by rejecting the
requirement of strict irreversibility. Higher to lower entropy
transitions are still allowed, but they are rendered unlikely, and
recurrence can be tamed by noting that the recurrence time for a
typical SM system is larger than age of the universe, which means that
we won't observe recurrence (Bricmont 1995; Callender 1999).
Yet, this amounts to admitting that entropy increase is not universal
and the formalism is compatible with there being periods of decreasing
entropy at some later point in the history of the universe.



A crucial ingredient of the Mentaculus is the Past-Hypothesis. The
idea of grounding thermodynamic behaviour in a cosmic low-entropy past
can be traced back to Boltzmann (Uffink 2007: 990) and has since been
advocated by prominent physicists like Feynman (1965: Ch. 5) and R.
Penrose (2004: Ch. 27). This raises two questions: first, can the
Past-Hypothesis be given a precise formulation that serves the purpose
of SM, and, second, what status does the Past-Hypothesis have and does
the fact that the universe started in this particular state require an
explanation?



As regards the first question, Earman has cast the damning verdict
that the Past-Hypothesis is "not even false" (2006)
because in cosmologies described in general relativity there is no
well-defined sense in which the Boltzmann entropy has a low value. A
further problem is that in the Mentaculus the Boltzmann entropy is a
global quantity characterising the entire universe. But, as Winsberg
points out, the fact that this quantity is low does not imply that the
entropy of a particular small subsystem of interest is also low, and,
worse just because the overall entropy of the universe increases it
need not be the case that the entropy in a small subsystem also
increases (2004a). The source of these difficulties is that the
Mentaculus takes the entire universe to be the relevant system and so
one might try get around them by reverting to where we started:
laboratory systems like gases in boxes. One can then take the
past state simply to be the state in which such a gas is prepared at
the beginning of a process (say in the left half of the container).
This leads to the so-called branch systems approach, because a system
is seen as "branching off" from the rest of the universe
when it is isolated from its environment and prepared in
non-equilibrium state (Davies 1974; Sklar 1993: 318-32). Albert
(2000) dismisses this option for a number of reasons, chief among them
that it is not clear why one should regard the statistical postulate
as valid for such a state (see Winsberg (2004b) for a discussion).



As regards the second question, Chen (forthcoming), Goldstein (2001),
and Loewer (2001) argue that Past-Hypothesis has the status of a
fundamental law of nature. Albert seems to regard it as something like
a Kantian regulative principle in that its truth must be assumed in
order to make knowledge of the past possible at all. By contrast,
Callender, Price, and Wald regard that the Past-Hypothesis a
contingent matter of fact, but they disagree on whether this fact
stands in need of an explanation. Price (1996, 2004) argues that it
does because the crucial question in SM is not why entropy increase,
but rather why it ever got to be low in the first place. Callender
(1998, 2004a, 2004b) disagrees: the Past-Hypothesis simply specifies
initial conditions of a process, and initial conditions are not the
kind of thing that needs to be explained (see also Sklar (1993:
309-18)). Parker (2005) argues that conditionalising on the
initial state of the universe does not have the explanatory power to
explain irreversible behaviour. Baras and Shenker (2020) and Farr
(2022) analysed the notion of explanation that is involved in this
debate and argue that different questions are in play that require
different answers.


### 4.7 The Long-Run Residence Time Account



The long-run residence time account offers a different perspective
both on the definition of equilibrium and the approach to it (Werndl
& Frigg 2015a, 2015b). Rather than first defining equilibrium
through combinatorial considerations (as in
 SS4.2)
 and *then* asking why systems approach equilibrium thus
defined (as do the accounts discussed in
 SSSS4.4-4.6),
 the long-run residence time account *defines* equilibrium
through thermodynamic behaviour. The account begins by characterising
the macro-states in the set \(\{ M\_{1},\ldots,M\_{n}\}\) in purely
macroscopic terms, i.e., through thermodynamic variables like pressure
and temperature, and then identifies the state in which a system
resides most of the time as the equilibrium state: among the
\(M\_{i}\), the equilibrium macro-state is by definition the state in
which a system spends most of its time in the long run (which gives
the account its name).



This definition requires no assumption about the size of the
equilibrium macro-region, but one can then show that it is a property
of the equilibrium macro-state that its macro-region is large. This
result is fully general in that it does not depend on assumptions like
particles being non-interacting (which makes it applicable to all
systems including liquids and solids), and it does not depend on
combinatorial considerations at the micro-level. The approach to
equilibrium is built into the definition in the sense that if there is
no macro-state in which the system spends most of its time, then the
system simply has no equilibrium. This raises the question of the
circumstances under which an equilibrium exists. The account answers
this question by providing a general existence theorem which furnishes
criteria for the existence of an equilibrium state (Werndl & Frigg
forthcoming-b). Intuitively, the existence theorem says that
there is an equilibrium just in case the system's state space is
split up into invariant regions on which the motion is ergodic and the
equilibrium macro-state is largest in size relative to the other
macro-states on each such region.



Like the account previously discussed, the long-run residence time
account accommodates Loschmidt's and Zermelo's objections
by rejecting the requirement of strict irreversibility: it insists
that being in equilibrium most of the time is as much as one can
reasonably ask for because actual physical systems show equilibrium
fluctuations and equilibrium is not the dead and immovable state that
thermodynamics says it is.


### 4.8 Problems and Limitations



BSM enjoys great popularity in foundational debates due to its clear
and intuitive theoretical structure. Nevertheless, BSM faces a number
of problems and limitations.



The first problem is that BSM only deals with closed systems that
evolve under their own internal dynamics. As we will see in
 Section 6,
 GSM successfully deals with systems that can exchange energy and even
particles with their environments, and systems of this kind play an
important role in SM. Those who think that SM only deals with the
entire universe can set this problem aside because the universe
(arguably) is a closed system. However, those who think that the
objects of study in SM are laboratory-size systems like gases and
crystals will have to address the issues of how BSM can accommodate
interactions between systems and their environments, which is a
largely ignored problem.



A second problem is that even though macro-states are ubiquitous in
discussions about BSM, little attention is paid to a precise
articulation of what these states are. There is loose talk about how a
system looks from macroscopic perspective, or there is a vague appeal
to thermodynamic variables. However, by the lights of thermodynamics,
variables like pressure and temperature are defined *only* in
equilibrium and it remains unclear how non-equilibrium states, and
with them the approach to equilibrium, should be characterised in
terms of thermodynamic variables. Frigg and Werndl (forthcoming-a)
suggest solving this problem by defining macro-states in terms of
local field-variables, but the issue needs further attention.



A third problem is that current formulations of BSM are closely tied
to deterministic classical systems
 (SS3).
 Some versions of BSM can be formulated based on classical stochastic
system (Werndl & Frigg 2017). But the crucial question is whether,
and if so how, a quantum version of BSM can be formulated (for a
discussion see the entry on
 quantum mechanics).
 Dizadji-Bahmani (2011) discusses how a result due to Linden and
others (2009) can be used to construct an argument for the conclusion
that an arbitrary small subsystem of a large quantum system typically
tends toward equilibrium. Chen (forthcoming) formulates a quantum
version of the Mentaculus, which he calls the *Wentaculus* (see
also his 2022). Goldstein, Lebowitz, Tumulka, and Zanghi (2020)
describe a quantum analogue of the Boltzmann entropy and argue that
the Boltzmannian conception of equilibrium is vindicated also in
quantum mechanics by recent work on thermalization of closed quantum
systems. These early steps have not yet resulted in comprehensive and
widely accepted formulation of quantum version of BSM, the formulation
of a such a version of remains an understudied topic. Albert (2000:
Ch. 7) suggested that the spontaneous collapses of the so-called GRW
theory (for introduction see the entry on
 collapse theories),
 a particular approach quantum mechanics, could be responsible for the
emergence of thermodynamic irreversibility. Te Vrugt, Toth and
Wittkowski (2021) put this proposal to test in computer simulations
and found that for initial conditions leading to anti-thermodynamic
behaviour GRW collapses do not lead to thermodynamic behaviour and
that therefore the GRW does not induce irreversible behaviour.



Finally, there is no way around recognising that BSM is mostly used in
foundational debates, but it is GSM that is the practitioner's
workhorse. When physicists have to carry out calculations and solve
problems, they usually turn to GSM which offers user-friendly
strategies that are absent in BSM. So either BSM has to be extended
with practical prescriptions, or it has to be connected to GSM so that
it can benefit from its computational methods (for a discussion of the
latter option see
 SS6.7).


## 5. The Boltzmann Equation



A different approach to the problem is taken by Boltzmann in his
famous (1872 [1966 Brush translation]) paper, which contains two
results that are now known as the Boltzmann Equation and the
*H*-theorem. As before, consider a gas, now described through a
distribution function \(f\_{t}(\vec{v})\), which specifies what
fraction of molecules in the gas has a certain velocity \(\vec{v}\) at
time \(t\). This distribution can change over time, and
Boltzmann's aim was to show that as time passes this
distribution function changes so that it approximates the
Maxwell-Boltzmann distribution, which, as we have seen in
 Section 4.2,
 is the equilibrium distribution for a gas.



To this end, Boltzmann derived an equation describing the time
evolution of \(f\_{t}(\vec{v})\). The derivation assumes that the gas
consists of particles of diameter \(D\) that interact like hard
spheres (i.e., they interact *only* when they collide); that
all collisions are elastic (i.e., no energy is lost); that the number
of particles is so large that their distribution, which in reality is
discrete, can be well approximated by a continuous and differentiable
function \(f\_{t}(\vec{v})\); and that the density of the gas is so low
that only two-particle collisions play a role in the evolution of
\(f\_{t}(\vec{v})\).



The crucial assumption in the argument is the so-called
"*Stosszahlansatz*", which specifies how many
collisions of a certain type take place in certain interval of time
(the German "*Stosszahlansatz*" literally means
something like "collision number assumption"). Assume the
gas has \(N\) molecules per unit volume and the molecules are equally
distributed in space. The type of collisions we are focussing on is
the one between a particle with velocity \(\vec{v}\_{1}\) and one with
velocity \(\vec{v}\_{2}\), and we want to know the number
\(N(\vec{v}\_{1}, \vec{v}\_{2})\) of such collisions during a small
interval of time \(\Delta t\). To solve this problem, we begin by
focussing on one molecule with \(\vec{v}\_{1}\). The relative velocity
of this molecule and a molecule moving with \(\vec{v}\_{2}\) is
\(\vec{v}\_{2} - \vec{v}\_{1}\) and the absolute value of that relative
velocity is \(\left\| \vec{v}\_{2} - \vec{v}\_{1} \right\|\). Molecules
of diameter D only collide if their centres come closer than \(D\). So
let us look at a cylinder with radius \(D\) and height \(\left\|
\vec{v}\_{2} - \vec{v}\_{1} \right\|\Delta t\), which is the volume in
space in which molecules with velocity \(\vec{v}\_{2}\) would collide
with our molecule during \(\Delta t\). The volume of this cylinder
is

 
\[\pi D^{2}\left\| \vec{v}\_{2} - \vec{v}\_{1} \right\|\Delta t .\]


If we now make the strong assumption that the initial velocities of
colliding particles are independent, it follows that number of
molecules with velocity \(\vec{v}\_{2}\) in a unit volume of the gas at
time \(t\) is \(Nf\_{t}(\vec{v}\_{2})\), and hence the number of such
molecules in our cylinder is

 
\[ N f\_{t} (\vec{v}\_{2}) \pi D^{2} \left\| \vec{v}\_{1} - \vec{v}\_{2} \right\| \Delta t.\]


This is the number of collisions that the molecule we are focussing on
can be expected to undergo during \(\Delta t\). But there is nothing
special about this molecule, and we are interested in the number of
*all* collisions between particles with velocities
\(\vec{v}\_{1}\) and \(\vec{v}\_{2}\). To get to that number, note that
the number of molecules with velocity \(\vec{v}\_{1}\) in a unit volume
of gas at time \(t\) is \(Nf\_{t}(\vec{v}\_{1})\). That is, there are
\(Nf\_{t}(\vec{v}\_{1})\) molecules like the one we were focussing on.
It is then clear that the total number of collisions can be expected
to be the product of the number of collisions for each molecule with
\(\vec{v}\_{1}\) times the number of molecules with
\(\vec{v}\_{1}\):

 
\[ N\left( \vec{v}\_{1}, \vec{v}\_{2} \right) = N^{2} f\_{t}(\vec{v}\_{1}) f\_{t}(\vec{v}\_{2}) \left\| \vec{v}\_{2} - \vec{v}\_{1} \right\| \pi D^{2}\Delta t. \]


This is the *Stosszahlansatz*. For ease of presentation, we
have made the mathematical simplification of treating
\(f\_{t}(\vec{v})\) as a fraction rather than as density in our
discussion of the *Stosszahlansatz*; for a statement of the
*Stosszahlansatz* for densities see, for instance, Uffink
(2007). Based on the *Stosszahlansatz*, Boltzmann derived what
is now known as the Boltzmann Equation:

 
\[\frac{\partial f\_{t}(\vec{v}\_{1})}{\partial t} = {\pi D^{2}N}^{2} \int\_{}^{} {d^{3}\vec{v}\_{2}} \left\| \vec{v}\_{2} - \vec{v}\_{1} \right\| \left( f\_{t} (\vec{v}\_{1}) f\_{t} (\vec{v}\_{2}) - f\_{t} (\vec{v}\_{1}^{\*}) f\_{t} (\vec{v}\_{2}^{\*}) \right),\]


where \(\vec{v}\_{1}^{\*}\) and \(\vec{v}\_{2}^{\*}\) are the velocities
of the particles *after* the collision. The integration is over
the space of the box that contains the gas. This is a so-called
integro-differential equation. The details of this equation need not
concern us (and the mathematics of such equations is rather tricky).
What matters is the overall structure, which says that the way the
density \(f\_{t}(\vec{v})\) changes over time depends on the difference
of the products of the densities of the incoming an of the outgoing
particles. Boltzmann then introduced the quantity \(H\),



\[H \left\lbrack f\_{t}(\vec{v}) \right\rbrack = \int\_{}^{} {d^{3}\vec{v}} f\_{t}(\vec{v}) \ln \left(f\_{t}(\vec{v})\right), \]


and proved that \(H\) decreases monotonically in time,

 
\[\frac{dH\left\lbrack f\_{t}(\vec{v}) \right\rbrack}{dt} \leq 0,\]


and that \(H\) is stationary (i.e., \(dH\lbrack f\_{t}(\vec{v})
\rbrack/dt = 0\)) iff \(f\_{t}(\vec{v})\) is the Maxwell-Boltzmann
distribution. These two results are the *H*-Theorem.



The definition of \(H\) bears formal similarities both to the
expression of the Boltzmann entropy in the combinatorial argument
 (SS4.3)
 and, as we will see, to the Gibbs entropy
 (SS6.3);
 in fact \(H\) looks like a negative entropy. For this reason the
*H*-theorem is often paraphrased as showing that entropy
increases monotonically until the system reaches the equilibrium
distribution, which would provide a justification of thermodynamic
behaviour based on purely mechanical assumptions. Indeed, in his 1872
paper, Boltzmann himself regarded it as a rigorous general proof of
the Second Law of thermodynamics (Uffink 2007: 965; Klein 1973:
73).



The crucial conceptual questions at this point are: what exactly did
Boltzmann prove with the *H*-theorem? Under which conditions is
the Boltzmann Equation valid? And what role do the assumptions, in
particular, the *Stosszahlansatz* play in deriving it? The
discussion of these question started four years after the paper was
published, when Loschmidt put forward his reversibility objection
 (SS4.3).
 This objection implies that \(H\) must be able to increase as well as
decrease. Boltzmann's own response to Loschmidt's
challenge and the question of the scope of the *H*-theorem is a
matter of much debate; for discussions see, for instance, Brown,
Myrvold, and Uffink (2009), Cercignani (1998), Brush (1976), and
Uffink (2007). We cannot pursue this matter here, but the gist of
Boltzmann's reply would seem to have been that he admitted that
there exists initial states for which \(H\) decreases, but that these
rarely, if ever, occur in nature. This leads to what is now known as a
statistical reading of the *H*-theorem: the *H*-theorem
shows entropy increase to be likely rather universal.



A century later, Lanford published a string of papers (1973, 1975,
1976, 1981) culminating in what is now known as Lanford's
theorem, which provides rigorous results concerning the validity of
the Boltzmann Equation. Lanford's starting point is the question
whether, and if so in what sense, the Boltzmann equation is consistent
with the underlying Hamiltonian dynamics. To this end, note that every
point \(x\) in the state space \(X\) of a gas has a distribution
\(f\_{x}(\vec{r}, \vec{v})\) associated with it, where \(\vec{r}\) and
\(\vec{v}\) are, respectively, the location and velocity of one
particle (recall from
 SS3
 that \(X\) contains the position and momenta of all molecules). For a
finite number of particles \(f\_{x}(\vec{r}, \vec{v})\) is not
continuous, let alone differentiable. So as a first step, Lanford
developed a way to obtain a differentiable distribution function
distribution \(f^{(x)}(\vec{r}, \vec{v})\), which involves taking the
so-called Boltzmann-Grad limit. He then evolved this distribution
forward in time both under the fundamental Hamiltonian dynamics, which
yields \(f\_{\text{Ht}}^{(x)}(\vec{r}, \vec{v})\), and under the
Boltzmann Equation, which yields \(f\_{\text{Bt}}^{(x)}(\vec{r},
\vec{v})\). Lanford's theorem compares these two distributions
and essentially says that for most points \(x\) in \(X\),
\(f\_{\text{Ht}}^{(x)}(\vec{r}, \vec{v})\) and
\(f\_{\text{Bt}}^{(x)}(\vec{r}, \vec{v})\) are close to each other for
times in the interval \(\left\lbrack 0, t^{\*} \right\rbrack,\) where
\(t^{\*}\) is a cut-off time (where "most" is judged by the
so-called microcanonical measure on the phase space; for discussion of
this measure see
 SS6.1).
 For rigorous statements and further discussions of the theorem see
Ardourel (2017), Uffink and Valente (2015), and Valente (2014).



Lanford's theorem is a remarkable achievement because it shows that a
statistical and approximate version of the Bolzmann Equation can be
derived from the Hamiltonian mechanics and most initial conditions in
the Bolzmann-Grad limit for a finite amount of time. In this sense it
can be seen as a vindication of Boltzmann's statistical version
of the *H*-theorem. At the same time the theorem also
highlights the limitations of the approach. The relevant distributions
are close to each other only up to time \(t^{\*}\), and it turns out
that \(t^{\*}\) is roughly two fifths of the mean time a particle moves
freely between two collisions. But this is a very short time! During
the interval \(\left\lbrack 0, t^{\*} \right\rbrack\), which for a gas
like air at room temperature is in the order of microseconds, on
average 40% of the molecules in the gas will have been involved in one
collision and the other 60% will have moved freely. This is patiently
too short to understand macroscopic phenomena like the one that we
described at the beginning of this article, which take place on a
longer timescale and will involve many collisions for all particles.
And like Boltzmann's original results, Lanford's theorem also depends
on strong assumptions, in particular a measure-theoretic version of
the *Stosszahlansatz* and Valente (cf. Uffink & Valente
2015).



Finally, one of the main conceptual problems concerning
Lanford's theorem is where the apparent irreversibility comes
from. Various opinions have been expressed on this issue. Lanford
himself first argued that irreversibility results from passing to the
Boltzmann-Grad limit (Lanford 1975: 110), but later changed his mind
and argued that the *Stosszahlansatz* for incoming collision
points is responsible for the irreversible behaviour (1976, 1981).
Cercignani, Illner, and Pulvirenti (1994) and Cercignani (2008) claim
that irreversibility arises as a consequence of assuming a hard-sphere
dynamics. Valente (2014) and Uffink and Valente (2015) argue that
there is no genuine irreversibility in the theorem because the theorem
is time-reversal invariant. For further discussions on the role of
irreversibility in Lanford's theorem, see also Lebowitz (1983),
Spohn (1980, 1991), and Weaver (2021, 2022)


## 6. Gibbsian Statistical Mechanics (GSM)



Gibbsian Statistical Mechanics (GSM) is an umbrella term covering a
number of positions that take Gibbs' (1902 [1981]) as their
point of departure. In this section, we introduce the framework and
discuss different articulations of it along with the issues they
face.


### 6.1 The Framework of GSM



Like BSM, GSM departs from the dynamical system \((X,\) \(\phi,\)
\(\mu)\) introduced in
 Section 3
 (although, as we will see below, it readily generalises to quantum
mechanics). But this is where the commonalities end. Rather than
partitioning \(X\) into macro-regions, GSM puts a probability density
function \(\rho(x)\) on \(X\), often referred to as a
"distribution". This distribution evolves under the
dynamics of the system through the law

 
\[\rho\_{t}(x) = \rho\_{0}(\phi\_{- t}(x))\]


where \(\rho\_{0}\) is the distribution the initial time \(t\_{0}\) and
\(\phi\_{- t}(x)\) is the micro-state that evolves into \(x\) during
\(t\). A distribution is called stationary if it does not change over
time, i.e., \(\rho\_{t}(x)= \rho\_{0}(x)\) for all \(t\). If the
distribution is stationary, Gibbs says that the system is in
"statistical equilibrium".



At the macro-level, a system is characterised by macro-variables,
which are functions \(f:X\rightarrow \mathbb{R}\), where
\(\mathbb{R}\) are the real numbers. With the exception of entropy and
temperature (to which we turn below), GSM takes all physical
quantities to be represented by such functions. The so-called
*phase average* of \(f\) is

 
\[\left\langle f \right\rangle = \int\_{X} f(x) \rho(x) dx.\]


The question now is how to interpret this formalism. The standard
interpretation is in terms of what is known as an ensemble. An
*ensemble* is an infinite collection of systems of the same
kind that differ in their state. Crucially, this is a collection of
copies of the *entire* system and not a collection of
molecules. For this reason, Schrodinger characterised an ensemble
as a collection of "mental copies of the one system under
consideration" (1952 [1989: 3]). Hence the members of an
ensemble do not interact with each other; an ensemble is not a
physical object; and ensembles have no spatiotemporal existence. The
distribution can then be interpreted as specifying "how
many" systems in the ensemble have their state in certain region
\(R\) of \(X\) at time \(t\). More precisely, \(\rho\_{t}(x)\) is
interpreted as giving the probability of finding a system in \(R\) at
\(t\) when drawing a system randomly from the ensemble in much the
same way in which one draws a ball from an urn:

 
\[p\_{t}(R) = \int\_{R} {\rho\_{t}(x)} dx.\]


What is the right distribution for a given physical situation? Gibbs
discusses this problem at length and formulates three distributions
which are still used today: the *microcanonical distribution*
for isolated systems, the *canonical distribution* for system
with fluctuating energy, and the *grand-canonical distribution*
for systems with both fluctuating energy and fluctuating particle
number. For a discussion of the formal aspects of these distributions
see, for instance, Tolman (1938 [1979]), and for philosophical
discussions see Davey (2008, 2009) and Myrvold (2016).



Gibbs' statistical equilibrium is a condition on an
*ensemble* being in equilibrium, which is different from an
*individual system* being in equilibrium (as introduced in
 SS1).
 The question is how the two relate, and what an experimenter who
measures a physical quantity on a system observes. A standard answer
one finds in SM textbooks appeals to the *averaging principle*:
when measuring the quantity \(f\) on a system in thermal equilibrium,
the observed equilibrium value of the property is the ensemble average
\(\langle f\rangle\) of an ensemble in ensemble-equilibrium. The
practice of applying this principle is often called *phase
averaging*. One of the core challenges for GSM is to justify this
principle.


### 6.2 Equilibrium: Why Does Phase Averaging Work?



The standard justification of phase averaging that one finds in many
textbooks is based on the notion of ergodicity that we have already
encountered in
 Section 4.4.
 In the current context, we consider the *infinite time
average* \(f^{\*}\)of the function \(f\). It is a mathematical fact
that ergodicity as defined earlier is equivalent to it being the case
that \(f^{\*} = \langle f \rangle\) for almost all initial states. This
is reported to provide a justification for phase averaging as follows.
Assume we carry out a measurement of the physical quantity represented
by \(f\). It will take some time to carry out the measurement, and so
what the measurement device registers is the time average over the
duration of the measurement. Indeed, the time needed to make the
measurement is long compared to the time scale on which typical
molecular processes take place, the measured result is approximately
equal to the infinite time average \(f^{\*}\). By ergodicity, \(f^{\*}\)
is equal to \(\langle f\rangle\), which justifies the averaging
principle.



This argument fails for several reasons (Malament & Zabell 1980;
Sklar 1993: 176-9). First, from the fact that measurements take
time it does not follow that what is measured are time averages, and
even if one could argue that measurement devices output time averages,
these would be *finite* time averages and equating these finite
time averages with infinite time averages is problematic because
finite and infinite averages can assume very different values even if
the duration of the finite measurement is very long. Second, this
account makes a mystery of how we observe change. As we have seen in
 Section 1,
 we do observe how systems approach equilibrium, and in doing so we
observe macro-variables changing their values. If measurements
produced infinite time averages, then no change would ever be observed
because these averages are constant. Third, as we already noted
earlier, ergodicity is a stringent condition and many systems to which
SM is successfully applied are not ergodic (Earman & Redei
1996), which makes equating time averages and phase averages
wrong.



A number of approaches have been designed to either solve or
circumvent these problems. Malament and Zabell (1980) suggest a method
of justifying phase averaging that still invokes ergodicity but avoids
an appeal to time averages. Vranas (1998) offers a reformulation of
this argument for systems that are epsilon-ergodic (see
 SS4.4).
 This accounts for systems that are "almost" ergodic, but
remains silent about systems that are far from being ergodic. Khinchin
(1949) restricts attention to systems with a large number of degrees
of freedom and so-called sum functions (i.e., functions that can are a
sum over one-particle functions), and shows that for such systems
\(f^{\*} = \langle f\rangle\) holds on the largest part of \(X\); for a
discussion of this approach see Batterman (1998) and Badino (2006).
However, as Khinchin himself notes, the focus on sum-functions is too
restrictive to cover realistic systems, and the approach also has to
revert to the implausible posit that observations yield infinite time
averages. This led to a research programme now known as the
"thermodynamic limit", aiming to prove
"Khinchin-like" results under more realistic assumptions.
Classic statements are Ruelle (1969, 2004); for a survey and further
references see Uffink (2007: 1020-8).



A different approach to the problem insists that one should take the
status of \(\rho(x)\) as a probability seriously and seek a
justification of averaging in statistical terms. In this vein, Wallace
(2015) insists that the quantitative content of statistical mechanics
is exhausted by the statistics of observables (their expectation
values, variances, and so on) and McCoy (2020) submits that
\(\rho(x)\) is the complete physical state of an individual
statistical mechanical system. Such a view renounces the association
of measurement outcomes with phase averages and insists that
measurements are "an instantaneous act, like taking a
snapshot" (O. Penrose 1970: 17-18): if a measurement of
the quantity associated with \(f\) is performed on a system at time
\(t\) and the system's micro-state at time \(t\) is \(x(t)\),
then the measurement outcome at time \(t\) will be \(f(x(t))\). An
obvious consequence of this definition is that measurements at
different times can have different outcomes, and the values of
macro-variables can change over time. One can then look at how these
values change over time. One way of doing this is to look at
fluctuations away from the average:

 
\[\Delta(t) = f\left( x(t) \right) - \left\langle f \right\rangle,\]


where \(\Delta(t)\) is the fluctuation away from the average at time
\(t\). One can then expect that a that the outcome of a measurement
will be \(\langle f\rangle\) if fluctuations turn out to be small and
infrequent. Although this would not seem to be the received textbook
position, something like it can be identified in some, for instance
Hill (1956 [1987]) and Schrodinger (1952 [1989]). A precise
articulation will have to use \(\rho\) to calculate the probability of
fluctuations of a certain size, and this requires the system to meet
stringent dynamical conditions, namely either the masking condition
or the *f*-independence condition (Frigg & Werndl
2021).


### 6.3 GSM and Approach to Equilibrium



As discussed so far, GSM is an equilibrium theory, and this is also
how it is mostly used in applications. Nevertheless, a comprehensive
theory of SM must also account for the approach to equilibrium. To
discuss the approach to equilibrium, it is common to introduce the
*Gibbs entropy*

 
\[S\_{G} = - k\int\_{X} \rho(x)\log\lbrack\rho(x)\rbrack dx.\]


The Gibbs entropy is a property of an ensemble characterised by a
distribution \(\rho\). One might then try to characterise the approach
to equilibrium as a process in which \(S\_{G}\) increases monotonically
to finally reach a maximum in equilibrium. But this idea is undercut
immediately by a mathematical theorem saying that \(S\_{G}\) is a
constant of motion:

 
\[{S\_{G}\lbrack\rho}\_{t}(x)\rbrack = S\_{G}\lbrack\rho\_{0}(x)\rbrack\]


for all times \(t\). So not only does \(S\_{G}\) fail to increase
monotonically; it does not change at all! This precludes a
characterisation of the approach to equilibrium in terms of increasing
Gibbs entropy. Hence, either such a characterisation has to be
abandoned, or the formalism has to be modified to allow \(S\_{G}\) to
increase.



A second problem is a consequence of the Gibbsian definition of
statistical equilibrium. As we have seen in
 SS6.1,
 a system is in statistical equilibrium if \(\rho\) is stationary. A
system away from equilibrium would then have to be associated with a
non-stationary distribution and eventually evolve into the stationary
equilibrium distribution. But this is mathematically impossible. It is
a consequence of the theory's formalism of GSM that a
distribution that is stationary at some point in time has to be
stationary at all times (past and future), and that a distribution
that is non-stationary at some point in time will always be
non-stationary. So an ensemble cannot evolve from non-stationary
distribution to stationary distribution. This requires either a change
in the definition of equilibrium, or a change in the formalism that
would allow distributions to change in requisite way.



In what follows we discuss the main attempts to address these
problems. For alternative approaches that we cannot cover here see
Frigg (2008b: 166-68) and references therein.


### 6.4 Coarse-Graining



Gibbs was aware of the problems with the approach to equilibrium and
proposed coarse-graining as a solution (Gibbs 1902 [1981]: Ch. 12).
This notion has since been endorsed by many practitioners (see, for
instance, Farquhar 1964 and O. Penrose 1970). We have already
encountered coarse-graining in
 SS4.2.
 The use of it here is different, though, because we are now
putting a grid on the full state space \(X\) and not just on the
one-particle space. One can then define a coarse-grained density
\(\bar{\rho}\) by saying that at every point \(x\) in \(X\) the
value of \(\bar{\rho}\) is the average of \(\rho\) over the grid cell
in which \(x\) lies. The advantage of coarse-graining is that the
coarse-grained distribution is not subject to the same limitations as
the original distribution. Specifically, let us call the Gibbs entropy
that is calculated with the coarse-grained distribution the
coarse-grained Gibbs entropy. It now turns out that coarse-grained
Gibbs entropy is not a constant of motion and it is possible for the
entropy to increase. This re-opens the avenue of understanding the
approach to equilibrium in terms of an increase of the entropy. It is
also possible for the coarse-grained distribution to evolve so that it
is spread out evenly over the entire available space and thereby comes
to look like a micro-canonical equilibrium distribution. Such a
distribution is also known as the quasi-equilibrium equilibrium
distribution (Blatt 1959; Ridderbos 2002).



Coarse-graining raises two questions. First, the coarse-grained
entropy *can* increase and the system *can* approach a
coarse-grained equilibrium, but under what circumstances will it
actually do so? Second, is it legitimate to replace standard
equilibrium by quasi-equilibrium?



As regards the first question, the standard answer (which also goes
back to Gibbs) is that the system has to be *mixing*.
Intuitively speaking, a system is mixing if every subset of \(X\) ends
up being spread out evenly over the entire state space in the long run
(for a more detailed account of mixing see entry on
 the ergodic hierarchy).
 The problem is that mixing is a very demanding condition. In fact,
being mixing implies being ergodic (because mixing is strictly
stronger than ergodicity). As we have already noticed, many relevant
systems are not ergodic, and hence *a fortiori* not mixing.
Even if a system is mixing, the mixed state is only achieved in the
limit for \(t \rightarrow \infty\), but real physical systems reach
equilibrium in finite time (indeed, in most cases rather quickly).



As regards the second question, the first point to note is that a
silent shift has occurred: Gibbs initially defined equilibrium through
stationarity while the above argument defines it through uniformity.
This needs further justification, but in principle there would seem to
be nothing to stop us from redefining equilibrium in this way.



The motivation for adopting quasi-equilibrium is that \(\bar{\rho}\)
and \(\rho\) are empirically indistinguishable. If the size of the
grid is below the measurement precision, no measurement will be able
to tell the difference between the two, and phase averages calculated
with the two distributions agree. Hence, hence there is no reason to
prefer \(\rho\) to \(\bar{\rho}\).



This premise has been challenged. Blatt (1959) and Ridderbos and
Redhead (1998) argue that this is wrong because the spin-echo
experiment (Hahn 1950) makes it possible to empirically discern
between \(\rho\) and \(\bar{\rho}\). The weight of this experiment
continues to be discussed controversially, with some authors insisting
that it invalidates the coarse gaining approach (Ridderbos 2002) and
others insisting that coarse-graining can still be defended (Ainsworth
2005; Lavis 2004; Robertson 2020). For further discussion see Myrvold
(2020b).


### 6.5 Interventionism



The approaches we discussed so far assume that systems are isolated.
This is an idealising assumption because real physical systems are not
perfectly isolated from their environment. This is the starting point
for the interventionist programme, which is based on the idea that
real systems are constantly subject to outside perturbations, and that
it is exactly these perturbations that drive the system into
equilibrium. In other words, it's these interventions from
outside the system that are responsible for its approach to
equilibrium, which is what earns the position the name
*interventionism*. This position has been formulated by Blatt
(1959) and further developed by Ridderbos and Redhead (1998). The key
insight behind the approach is that two challenges introduced in
 Section 6.3
 vanish once the system is not assumed to be isolated: the entropy can
increase, and a non-stationary distribution can be pushed toward a
distribution that is stationary in the future.



This approach accepts that isolated systems do not approach
equilibrium, and critics wonder why this would be the case. If one
places a gas like the one we discussed in
 Section 1
 somewhere in interstellar space where it is isolated from outside
influences, will it really sit there confined to the left half of the
container and not spread? And even if this were the case, would adding
just *any* environment resolve the issue? Interventionist
sometimes seem to suggest that this is the case, but in an unqualified
form this claim cannot be right. Environments can be of very different
kinds and there is no general theorem that says that any environment
drives a system to equilibrium. Indeed, there are reasons to assume
that there is no such theorem because while environments do drive
systems, they need not drive them to equilibrium. So it remains an
unresolved question under what conditions environments drive systems
to equilibrium.



Another challenge for interventionism is that one is always free to
consider a larger system, consisting of our original system plus its
environment. For instance, we can consider the "gas + box"
system. This system would then also approach equilibrium because of
outside influences, and we can then again form an even larger system.
So we get into a regress that only ends once the system under study is
the entire universe. But the universe has no environment that could
serve as a source of perturbations which, so the criticism goes, shows
that the programme fails.



Whether one sees this criticism as decisive depends on one's
views of laws of nature. The argument relies on the premise that the
underlying theory is a universal theory, i.e., one that applies to
everything that there is without restrictions. The reader can find an
extensive discussion in the entry on
 laws of nature.
 At this point we just note that while universality is widely held,
some have argued against it because laws are always tested in highly
artificial situations. Claiming that they equally apply outside these
settings involves an inductive leap that is problematic; see for
instance Cartwright (1999) for a discussion of such a view. This, if
true, successfully undercuts the above argument against
interventionism.


### 6.6 The Epistemic Account



The epistemic account urges a radical reconceptualization of SM. The
account goes back to Tolman (1938 [1979]) and has been brought to
prominence by Jaynes in a string of publications between 1955 and
1980, most of which are gathered in Jaynes (1983). On this approach,
SM is about our *knowledge* of the world and not about the
world itself, and the probability distributions in GSM represents our
state of knowledge about a system and not some matter of fact. The
centre piece of this interpretation is the fact that the Gibbs entropy
is formally identical to the Shannon entropy in information theory,
which is a measure for the lack of information about a system: the
higher the entropy, the less we know (for a discussion of the Shannon
entropy see the entry on
 information, SS4.2).
 The Gibbs entropy can therefore be seen as quantifying our lack of
information about a system. This has the advantage that ensembles are
no longer needed in the statement of GSM. On the epistemic account,
there is only one system, the one on which we are performing our
experiments, and \(\rho\) describes what we know about it. This also
offers a natural criterion for identifying equilibrium distributions:
they are the distributions with the highest entropy consistent with
the external constraints on the system because such distributions are
the least committal distributions. This explains why we expect
equilibrium to be associated with maximum entropy. This is known as
Jaynes' *maximum entropy principle* (MEP).



MEP has been discussed controversially, and, to date, there is no
consensus on its significance, or even cogency. For discussions see,
for instance, Denbigh and Denbigh (1985), Howson and Urbach (2006),
Lavis (1977), Lavis and Milligan (1985), Seidenfeld (1986), Shimony
(1985), Uffink (1995, 1996a), and Williamson (2010). The epistemic
approach also assumes that experimental outcomes correspond to phase
averages, but as we have seen, this is a problematic assumption
 (SS6.1).
 A further concern is that the system's own dynamics plays no
role in the epistemic approach. This is problematic because if the
dynamics has invariant quantities, a system cannot access certain
parts of the state space even though \(\rho\) may assign a non-zero
probability to it (Sklar 1993: 193-4).



The epistemic account's explanation of the approach to
equilibrium relies on making repeated measurements and
conditionalizing on each measurement result; for a discussion see
Sklar (1993: 255-257). This successfully gets around the problem
that the Gibbs entropy is constant, because the value assignments now
depend not only on the system's internal dynamics, but also on
the action of an experimenter. The problem with this solution is that
depending on how exactly the calculations are done, either the entropy
increase fails to be monotonic (indeed entropy decreases are possible)
or the entropy curve will become dependent on the sequence of instants
of time chosen to carry out measurements (Lavis & Milligan
1985).



However, the most fundamental worry about the epistemic approach is
that it fails to realise the fundamental aim of SM, namely to explain
how and why processes in nature take place because these processes
cannot possibly depend on what we know about them. Surely, so the
argument goes, the boiling of kettles or the spreading of gases has
something to do with how the molecules constituting these systems
behave and not with what we happen (or fail) to know about them
(Redhead 1995; Albert 2000; Loewer 2001). For further discussions of
the epistemic approach see Anta (forthcoming-a, forthcoming-b),
Shenker (2020), and Uffink (2011).


### 6.7 The Relation between GSM and BSM



A pressing and yet understudied question in the philosophy of SM
concerns the relation between the GSM and BSM. GSM provides the tools
and methods to carry out a wide range of equilibrium calculations, and
it is the approach predominantly used by practitioners in the field.
Without it, the discipline of SM would not be able to operate (Wallace
2020). BSM is conceptually neat and is preferred by philosophers when
they give foundational accounts of SM. So what we're facing is a
schism whereby the day-to-day work of physicists is in one framework
and foundational accounts and explanations are given in another
framework (Anta 2021a). This would not be worrisome if the frameworks
were equivalent, or at least inter-translatable in relatively clear
way. As the discussion in the previous sections has made clear, this
is not the case. And what is more, in some contexts the formalisms do
not even give empirically equivalent predictions (Werndl & Frigg
2020b). This raises the question of how exactly the two approaches are
related. Lavis (2005) proposes a reconciliation of the two frameworks
through giving up on the binary property of the system being or not
being in equilibrium, which should be replaced by the continuous
property of *commonness*. Wallace (2020) argues that GSM is a
more general framework in which the Boltzmannian approach may be
understood as a special case. Frigg and Werndl suggest that BSM is a
fundamental theory and GSM is an effective theory that offers means to
calculate values defined in BSM (Frigg & Werndl 2019; Werndl &
Frigg 2020a). Goldstein (2019) plays down their difference and argues
that the conflict between them is not as great as often imagined.
Finally, Goldstein, Lebowitz, Tumulka, and Zanghi (2020)
compare the Boltzmann entropy and the Gibbs entropy and argue that the
two notions yield the same (leading order) values for the entropy of a
macroscopic system in thermal equilibrium.


## 7. Further Issues



So far we have focussed on the questions that arise in the
articulation of the theory itself. In this section we discuss some
further issue that arise in connection with SM, explicitly excluding a
discussion of the direction of time and other temporal asymmetries,
which have their own entry in this encyclopedia (see the entry on
 thermodynamic asymmetry in time).


### 7.1 The Interpretation of SM Probabilities



How to interpret probabilities is a problem with a long philosophical
tradition (for a survey of different views see the entry on
 interpretations of probability).
 Since SM introduces probabilities, there is a question of how these
probabilities should be interpreted. This problem is particularly
pressing in SM because, as we have seen, the underlying mechanical
laws are deterministic. This is not a problem so long as the
probabilities are interpreted epistemically as in Jaynes'
account
 (SS6.6).
 But, as we have seen, a subjective interpretation seems to clash with
the realist intuition that SM is a physical theory that tells us how
things are independently of what we happen to know about them. This
requires probabilities to be objective.



Approaches to SM that rely on ergodic theory tend to interpret
probabilities as time-averages, which is natural because ergodicity
provides such averages. However, long-run time averages are not a good
indicator for how a system behaves because, as we have seen, they are
constant and so do not indicate how a system behaves out of
equilibrium. Furthermore, interpreting long-run time averages as
probabilities is motivated by the fact the that these averages seem to
be close cousins of long-run relative frequencies. But this
association is problematic for a number of reasons (Emch 2005;
Guttmann 1999; van Lith 2003; von Plato 1981, 1982, 1988,
1994). An alternative is to interpret SM probabilities as
propensities, but many regard this as problematic because
propensities would ultimately seem to be incompatible with a
deterministic underlying micro theory (Clark 2001).



Loewer (2001) suggested that we interpret SM probabilities as Humean
objective chances in Lewis' sense (1980) because the Mentaculus
(see
 SS4.6)
 is a best system in Lewis' sense. Frigg (2008a) identifies some
problems with this interpretation, and Frigg and Hoefer (2015)
formulate an alternative Humean account that is designed to overcome
these issues. For further discussion of Humean chances in SM, see
Beisbart (2014), Dardashti, Glynn, Thebault, and Frisch (2014),
Hemmo and Shenker (2022),  Hoefer (2019), and Myrvold (2016,
2021).


### 7.2 Maxwell's Demon and the Entropy Costs of Computation



Consider the following scenario, which originates in a letter that
Maxwell wrote in 1867 (see Knott 1911). Recall the vessel with a
partition wall that we have encountered in
 Section 1,
 but vary the setup slightly: rather than having one side empty, the
two sides of the vessel are filled with gases of different
temperatures. Additionally, there is now a shutter in the wall which
is operated by a demon. The demon carefully observes all the
molecules. Whenever a particle in the cooler side moves towards the
shutter the demon checks its velocity, and if the velocity of the
particle is greater than the mean velocity of the particles on the
hotter side of the vessel he opens the shutter and lets the particle
pass through to the hotter side. The net effect of the demon's
actions is that the hotter gas becomes even hotter and that the colder
gas becomes even colder. This means that there is a heat transfer from
the cooler to the hotter gas without doing any work because the heat
transfer is solely due to the demon's skill and intelligence in
sorting the molecules. Yet, according to the Second Law of
thermodynamics, this sort of heat transfer is not allowed. So we
arrive at the conclusion that the demons' action result in a
violation of the Second Law of thermodynamics.



Maxwell interpreted this scenario as a thought experiment that showed
that the Second Law of thermodynamics is not an exceptionless law and
that it has only "statistical certainty" (see Knott 1911;
Hemmo & Shenker 2010). Maxwell's demon has given rise to a
vast literature, some of it in prestigious physics journals. Much of
this literature has focused on exorcising the demon, i.e., on showing
that a demon would not be physically possible. Broadly speaking, there
are two approaches. The first approach is commonly attributed to
Szilard (1929 [1990]), but also goes also back to von Neumann (1932
[1955]) and Brillouin (1951 [1990]). The core idea of this approach is
that gaining information that allows us to distinguish between \(n\)
equally likely states comes at a necessary minimum cost in
thermodynamic entropy of \(k \log(n)\), which is the entropy
dissipated by the system that gains information. Since the demon has
to gain information to decide whether to open the shutter, the second
law of thermodynamics is not violated. The second approach is based on
what is now called *Landauer's principle*, which states
that in erasing information that can discern between \(n\) states, a
minimum thermodynamic entropy of \(k \log(n)\) is dissipated (Landauer
1961 [1990]). Proponents of the principle argue that because a demon
has to erase information on memory devices, Landauer's principle
prohibits a violation of the second law of thermodynamics.



In two influential articles Earman and Norton (1998, 1999) lament that
from the point of view of philosophy of science the literature on
exorcising the demon lacks rigour and reflection on what the goals the
enterprise are, and that the demon has been discussed from various
different perspectives, often leading to confusion. Earman and Norton
argue that the appeal to information theory has not resulted in a
decisive exorcism of Maxwell's demon. They pose a dilemma for
the proponent of an information theoretic exorcism of Maxwell's
demon. Either the combined system of the vessel and the demon are
already assumed to be subject to the second law of thermodynamics, in
which case it is trivial that the demon will fail. Or, if this is not
assumed, then proponents of the information theoretic exorcism have to
supply new physical principles to guarantee the failure of the demon
and they have to give independent grounds for it. Yet, in Earman and
Norton's view, such independent grounds have not been
convincingly established.



Bub (2001) and Bennett (2003) responded to Earman and Norton that if
one assumes that the demon is subject to the Second Law of
thermodynamics, the merit of Landauer's principle is that it
shows where the thermodynamic costs arise. Norton (2005, 2017) replies
that no general precise principle is stated how erasure and the
merging of computational paths necessarily lead to an increase in
thermodynamic entropy. He concludes that the literature on
Landauer's principle is too fragile and too tied to a few
specific examples to sustain general claims about the failure of
Maxwell's demons. Maroney (2005) argues that thermodynamic
entropy and information-theoretic entropy are conceptually different,
and that two widespread generalisations of Landauer's
fail. Maroney (2009) then formulates what he regards as a more precise
generalisation of Landauer's principle, which he argues does not
fail.



The discussions around Maxwell's demon are now so extensive that
they defy documentation in an introductory survey of SM. Classical
papers on the matter are collected in Leff and Rex (1990). For more
recent discussion see, for instance, Anta (2021b), Hemmo and Shenker
(2012; 2019), Ladyman and Robertson (2013, 2014), Leff and Rex (1994),
Myrvold (forthcoming), Norton (2013), and references therein.


### 7.3 The Gibbs Paradox



So far, we have considered how one gas evolves. Now let's look
at what happens when we mix two gases. Again, consider a container
with a partition wall in the middle, but now imagine that there are
two *different* gases on the left and on the right (for
instance helium and hydrogen) where both gases have the *same*
temperature. We now remove the shutter, and the gases start spreading
and get mixed. If we then calculate the entropy of the initial and the
final state of the two gases, we find that the entropy of the mixture
is greater than the entropy of the gases in their initial
compartments. This is the result that we expect. The paradox arises
from the fact that the calculations do *not* depend on the fact
that the gases are different: if we assume that we have air of the
same temperature on both sides of the barrier the calculations still
yield an increase in entropy when the barrier is removed. This seems
wrong because it would imply that the entropy of a gas depends on its
history and cannot be a function of its thermodynamic state alone (as
thermodynamics requires). This is known as the *Gibbs
Paradox*.



The standard textbook resolution of the paradox is that classical SM
gets the entropy wrong because it counts states that differ only by a
permutation of two indistinguishable particles as distinct, which is a
mistake (Huang 1963). So the problem is rooted in the notion of
individuality, which is seen as inherent to classical mechanics.
Therefore, so the argument goes, the problem is resolved by quantum
mechanics, which treats indistinguishable particles in the right way.
This argument raises a number of questions concerning the nature of
individuality in classical and quantum mechanics, the way of counting
states in both the Boltzmann and the Gibbs approach, and the relation
of SM to thermodynamics. Classical discussions include Denbigh and
Denbigh (1985: Ch. 4), Denbigh and Redhead(1989), Jaynes (1992),
Lande (1965), Rosen (1964), and van Kampen (1984). For more
recent discussions, see, for instance, Huggett (1999), Saunders
(2006), and Wills (forthcoming), as well as the contributions to Dieks
and Saunders (2018) and references therein.


### 7.4 SM Beyond Physics



Increasingly, the methods of SM are used to address problems outside
physics. Costantini and Garibaldi (2004) present a generalised version
of the Ehrenfest flea model and show that it can be used to describe a
wide class of stochastic processes, including problems in population
genetics and macroeconomics. Colombo and Palacios (2021) discuss the
application of the free energy principle in biology. The most prolific
application of SM methods outside physics are in economics and
finance, where an entire field is named after them, namely
econophysics. For discussions of different aspects of econophysics see
Jhun, Palacios, and Weatherall (2018); Kutner *et al*. (2019),
Rickles (2007, 2011), Schinckus (2018), Thebault, Bradley, and
Reutlinger (2017), and Voit (2005).


### 7.5 Reductionism and Inter-Theory Relations



In the introduction we said that SM had to account for the
thermodynamic behaviour of physical systems like gases in a box. This
is a minimal aim that everybody can agree on. But many would go
further and say that there must be stronger reductive relations
between SM and other parts of science, or, indeed, as in the
Mentaculs, that all layers of reality in the entire universe reduce to
SM. For a general discussion of reductionism and inter-theory
relations see the relevant entries in this encyclopaedia
 (scientific reduction
 and
 intertheory relations in physics).



A point where the issue of reduction has come to head is the
discussion of phase transitions. Consider again a container full of
gas and imagine that the container is over 100degC hot and that the
gas in it is water vapour. Now you start cooling down the container.
Once the temperature of the gas falls below 100degC, the gas
condenses and turns in liquid water, and once it falls below 0degC,
liquid water turns into solid ice. These are examples of *phase
transitions*, namely, first from the gaseous phase to the liquid
phase and then from the liquid to the solid phase. Thermodynamics
characterises phase transitions as a discontinuity in a thermodynamic
potential such as the free energy, and a phase transition is said to
occur when this potential shows a discontinuity (Callender 2001). It
now turns out that SM can reproduce this only in the so-called
thermodynamic limit, which takes the particle number in the system
toward infinity (while keeping the number of particles per volume
constant). This would seem to imply that phase transitions can only
occur in infinite systems. In this vein, physicist David Ruelle notes
that phase transitions can occur only in systems that are
"idealized to be actually infinite" (Ruelle 2004: 2). This
is sparked a heated debate over whether, and if so in what sense,
thermodynamics can be reduced to SM, because in nature phase
transitions clearly do occur in finite systems (the water in a puddle
freezes!). Batterman (2002) draws the conclusion that phase
transitions are *emergent* phenomena that are irreducible to
the underling micro-theory, namely SM. Others push back against this
view and argue that no actual infinity is needed and that therefore
limiting behaviour is neither emergent not indicative of a failure of
reduction. Norton (2012) reaches this conclusion by demoting the limit
to a mere approximation, Callender (2001) by counselling against
"taking thermodynamics too seriously", and Butterfield by
developing a view that reconciles reduction and emergence (2011a,
2011b, 2014). For further discussions of phase transitions and their
role in understanding whether, and if so how, thermodynamics can be
reduced to SM see, Ardourel (2018), Bangu (2009), Butterfield and
Buatta (2012), Franklin (2018), Liu (2001), Palacios (2018, 2019,
forthcoming), and Menon and Callender (2013), and for a discussion of
infinite idealisations in general Shech (2018).



Similar questions arise when we aim to reduce the thermodynamic
entropy to one of the SM entropies (Callender 1999; Dizadji-Bahmani et
al. 2010; Myrvold 2011), when we focus on the peculiar nature of
quasi-static processes in thermodynamics (Robertson forthcoming), when
we study equilibration Myrvold (2020a), and when we give up the
idealisation that SM systems are isolated and take gravity into
account (Callender 2011). There is also a question about what exactly
reduction is expected to achieve. Radical reductionists seem to expect
that once the fundamental level is sorted out, the rest of science
follows from it as a corollary (Weinberg 1992), which is also a vision
that seems to drive the Mentaculus. Others are more cautions. Lavis,
Kuhn, and Frigg (2021) and Yi (2003) note that even if reduction
is successful, thermodynamics remains in place as an independent
theory: SM requires the framework of thermodynamics, which serves as
recipient of information from SM but without itself being derivable
from SM.