{'fraction_non_alphanumeric': 0.04118194509073988, 'fraction_numerical': 0.007561656584457887, 'mean_word_length': 4.818274111675127, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The microcanonical thermodynamics of finite systems: The microscopic origin of condensation and phase separations; and the conditions for heat flow from lower to higher temperatures

Microcanonical thermodynamics allows the application of statistical mechanics both to finite and even small systems and also to the largest, self-gravitating ones. However, one must reconsider the fundamental principles of statistical mechanics especially its key quantity, entropy. Whereas in conventional thermostatistics, the homogeneity and extensivity of the system and the concavity of its entropy are central conditions, these fail for the systems considered here. For example, at phase separation, the entropy, S(E), is necessarily convex to make exp[S(E)-E/T] bimodal in E. Particularly, as inhomogeneities and surface effects cannot be scaled away, one must be careful with the standard arguments of splitting a system into two subsystems, or bringing two systems into thermal contact with energy or particle exchange. Not only the volume part of the entropy must be considered. As will be shown here, when removing constraints in regions of a negative heat capacity, the system may even relax under a flow of heat (energy) against a temperature slope. Thus the Clausius formulation of the second law: ``Heat always flows from hot to cold'', can be violated. Temperature is not a necessary or fundamental control parameter of thermostatistics. However, the second law is still satisfied and the total Boltzmann entropy increases. In the final sections of this paper, the general microscopic mechanism leading to condensation and to the convexity of the microcanonical entropy at phase separation is sketched. Also the microscopic conditions for the existence (or non-existence) of a critical end-point of the phase-separation are discussed. This is explained for the liquid-gas and the solid-liquid transition.

reconsider the fundamental principles of statistical mechanics especially its key quantity, entropy. Whereas in conventional thermostatistics, the homogeneity and extensivity of the system and the concavity of its entropy are central conditions, these fail for the systems considered here. For example, at phase separation, the entropy, S(E), is necessarily convex to make e S(E)-E/T bimodal in E. Particularly, as inhomogeneities and surface effects cannot be scaled away, one must be careful with the standard arguments of splitting a system into two subsystems, or bringing two systems into thermal contact with energy or particle exchange. Not only the volume part of the entropy must be considered; the addition of any other external constraint, [A. Wehrl, Rev. Mod. Phys. 50, 221 (1978)], such as a dividing surface, or the enforcement of gradients of the energy or particle profile, reduce the entropy. As will be shown here, when removing such constraints in regions of a negative heat capacity, the system may even relax under a flow of heat (energy) against a temperature slope. Thus the Clausius formulation of the second law: "Heat always flows from hot to cold," can be violated. Temperature is not a necessary or fundamental control parameter of thermostatistics. However, the second law is still satisfied and the total Boltzmann entropy increases. In the final sections of this paper, the general microscopic mechanism leading to condensation and to the convexity of the microcanonical entropy at phase separation is sketched. Also the microscopic conditions for the existence (or non-existence) of a critical end-point of the phase-separation are discussed. This is explained for the liquid-gas and the solid-liquid transition.

1.
The definition of entropy.
The internal energy, U, and the entropy, S, are the fundamental entities of thermodynamics. Therefore, proper understanding of them is essential.
Internal energy is defined through its changes, by the first law of thermodynamics, as the balancing entity between heat absorbed and external work performed by the system. Such definition is both intuitively clear and also consistent with a mechanistic perspective of the physical universe, and states the conservation of energy.
Entropy is defined through the two, crucially different, mechanisms of its changes: the heat exchanged externally by the system, inversely-weighted by the system temperature; and the internal production of entropy. The latter is fundamentally related to the spontaneous evolution ("Verwandlungen" 1 ) of the system, and states the universal irreversibility of spontaneous transitions. The relationship of entropy to the second law is often beset with confusion between external transfers of entropy and its internal production.
Because entropy is defined as an extensive variable, its change with time is determined by two distinct mechanisms: the flow of entropy to or from its defining volume (the thermodynamic system under consideration); and its internal production. Clausius originally called dQ' the change in "uncompensated heat;" for he understood that there is always in a thermodynamic transformation a quantity of heat evolved which is intrinsically unrelated to whatever heat that the system may absorb from or transfer to the rest of the universe, dQ. Clausius then went on to enunciate the second law as: In enunciating the second law of thermodynamics, Clausius quite clearly stated that, while the first type of entropy change (that effected by exchange of heat with its surroundings) can be positive, negative or zero, the second type of entropy change (that caused by the internal creation of entropy) can be only positive in any spontaneous transformation.
Boltzmann 3 later defined the entropy of an isolated system (for which dQ = 0) as an absolute, extensive, thermodynamic variable in terms of the sum of possible configurations, W, which the system can assume consistent with its thermodynamic constraints: Equation (3) remains the most fundamental and simple definition of entropy, and is engraved on Boltzmann's tombstone. This definition of entropy, (3), presciently acknowledges the absolute quality of entropy. Because W is a natural number, and ≥ 1, the inequality S ≥ 0 follows.
The understanding of entropy is sometimes obscured by frequent use of the Boltzmann-Gibbs canonical ensemble, and thermodynamic limit. In a semi-classical approximation, in which E is the total energy, N is the number of particles and V the volume. In equation (4), the density function, ρ (p,q), is that which defines the microcanonical ensemble: For a finite quantum-mechanical system: In equations (4) and (6), ε 0 represents the macroscopic energy resolution. In equations (3), (4), and (6), there is no need of the thermodynamic limit, nor of concavity, extensivity and homogeneity. In its semi-classical approximation, equation (4) using general densities, ρ, (which were not necessarily projection-like).
Wehrl discusses the conventional, canonical, Boltzmann-Gibbs statistics where all constraints are fixed only to their mean, allowing for free fluctuations. These free, unrestricted fluctuations of the energy imply an uncontrolled energy exchange with the universe, dQ in Clausius'  (3) and (6). We thus explore statistical mechanics and entropy at their most fundamental level. Because such analysis does not demand scaling or extensivity, it can further be applied to the much wider group of non-extensive systems from nuclei to galaxies 6 This statistical definition, (4) and (6) (called S surf in 11 ), has a priori not much to do with the adiabatic invariants of a single trajectory in the 6N-dimension. phase space.
As will be shown in section 3, the entropy, S(E), can be convex or concave with respect to E for non-extensive systems, -in contrast to the assumption made by Hertz. 10 In such cases, the temperature, or the mean kinetic energy per particle, does not control the energy flow during the equilibration of two systems in thermal contact, -as was the crucial argument of Hertz to introduce S bulk , (8). Contrastingly, the Boltzmann entropy, (4), still controls the direction of the energy flow under equilibration even in these somewhat counter-intuitive, though ubiquitous, situations. This clearly emphasizes that the Boltzmann entropy, (3), is the fundamental quantity for statistical mechanics, -not the temperature. (This statement refutes contradictory assertions made previously. [9][10][11] Moreover, the definition of the bulk entropy S bulk (E,N,V), (8), contains also energetically inaccessible states, characterized by less energy, which are clearly excluded by our knowledge of the total energy. Thus S bulk (E,N,V) does not measure the redundancy, or ignorance, of the available information, and is therefore unacceptable for conceptual reasons. This is a clear warning that one should not follow some formal definition and disregard the original clear meaning of entropy and statistics, as is too often done in the literature. Of course, for homogeneous macroscopic systems, the difference between the two alternative definitions of the entropy equations (4) and (6), and equation (8) disappears.

2.
The 0 th Law in conventional, extensive thermodynamics. 1 = E This section and the following discuss mainly systems that have no macroscopic (extensive) control parameters except energy; the particle density does not change, and there are no chemical reactions. In conventional (extensive) thermodynamics, the thermal equilibrium of two systems is established by bringing them into thermal contact and allowing exchange of free energy. Equilibrium is established when the total entropy, is maximal, such that Under an energy flux from 2→1, the total entropy changes to lowest order in by in which Consequently, a maximum of From here, Clausius' first formulation of the Second Law follows: "Heat always flows spontaneously from hot to cold." Essential for this conclusion is the additivity of S under the separation of the two subsystems, equation (9). Thus, temperature is an appropriate control parameter for extensive systems.

Stability against spontaneous energy gradients: There exists no phase separation without a convex, non-extensive entropy, S(E).
Small systems and very large self-gravitating systems are inhomogeneous, 12 as are essentially also systems at phase separation. For these cases, the additivity and extensivity of S, and also of E, do not hold. 12 The main purpose to develop this new and extended version of thermodynamics 12 is to address such systems. The Boltzmann-Planck entropy, equation (6), does not scale linearly with the volume or number of particles, but has an important, non-linear, correction which we shall call henceforth ∆S surf-corr .
Since the beginning of thermodynamics in the 19 th century, its principle motivation was to describe steam engines and the gas-liquid phase transition of water. In the gas-liquid phase transition, water becomes inhomogeneous and develops a separation of the gas phase from the liquid,i.e. it boils. As conventional canonical statistical mechanics works only for homogeneous, infinite systems, phase separations remain outside standard Boltzmann-Gibbs statistical thermodynamics and are identified by Yang-Lee singularities.
At phase separation in general, the microcanonical caloric curve T(E) is backbending.
Here the heat capacity, The weight, e (S(E)-E/T) , of the configurations with energy E in the definition of the canonical partition sum, becomes bimodal; at the transition temperature, it has two peaks, the liquid and the gas config-  (1). Consider the two equal systems depicted in Fig. 1 to be combined, such that one has the energy per lattice site e a = e 1 -∆e at temperature T a , and the other has energy per lattice site e b = e 3 +∆e at temperature T b ; where T b < T a . Consider also that energy is allowed to be exchanged without constraint. Then the systems will equilibrate at energy e 2 with an increase of entropy. The temperature T a drops(cooling) and energy, in the form of heat, flows (on the average) from b → a.
Thus heat flows spontaneously from the colder system to the hotter one; and the even though the difference of the first and the last term is positive.
Consider a transformation in the inverse direction: By relaxing the constraint and allowing, on average, for an energy flux, ∆E b→a > 0, against T a -T b > 0, (i.e., against the temperature gradient but in the direction of the decreasing energy gradient), an increase of S total → S 2 obtains.
This result is consistent with the normal picture of energy equilibration. Thus the Clausius stateent, "energy flows always from hot to cold," which is a statement of the dominant role of the temperature in thermo-statistics, 10 is violated.
These considerations show that, unlike extensive thermodynamics, the temperature is not an appropriate control parameter in non-extensive systems. In the thermodynamic limit of a system with short-range coupling, N→ ∞ as ∆S surf-corr ~ N 2/3 , and therefore (∆S surf-corr )/N ~ N -1/3 , must go to 0 in accordance with van Hove's theorem. Here following are illuminated the general microscopic mechanism leading to the appearance of a convex intruder in S(E,V,N,⋅⋅⋅). This is the generic signal of phase transitions of first order and of phase-separation within the microcanonical ensemble.

The microcanonical mechanism leading to condensation, phase separation, and the origin of the convexities of S(E).
Assume the system is classical and obeys the following Hamiltonian: In this case the system is controlled by energy and volume.

The gas-liquid phase transition.
The microcanonical sum of states or partition sum is: in which and V is the spatial volume; E 0 = min(Φ int ) is the energy of the ground-state of the system.
The separation of W(E,N,V) into W IG and W int is the microcanonical analogue of the split of the canonical partition sum into a kinetic part and a configuration part: In the thermodynamic limit, the order parameter of the (homogeneous) liquid-gas transition is the density. The transition is linked to a condensation of the system towards a larger density controlled by pressure. For a finite system, we expect analogous behavior. However, for a finite system, the transition is controlled by the constant system volume V. At low energies, the N particles condensate into a droplet with much smaller volume V 0,N . 3(N-1) internal coordinates are limited to V 0,N . Only the center of mass of the droplet can move freely in V (remember we did not fix the center-of-mass in equation (17)). The system does not fill the 3N-configuration space V N . Only a stripe with width (V 0N ) 1/3 in 3(N-1) dimensions of the total 3N-dimension space is populated. The system is non-homogeneous even though it is equilibrated and, at low energies, internally in the single liquid phase; and it is not characterized by an intensive homogeneous density. In fact, W int (E-E 0 ,N,V) can be written as: The first factor, , in equation (22) sorts the energetically forbidden region from the potential holes (clusters) in the 3N-dimension potential surface W int (E-E 0 ,N,V) and S int (E-E 0 ,N,V) have the limiting values: In the microcanonical ensemble this is analogue to the phenomenon of multi-fragmentation in nuclear systems. 12,21 This, in contrast to the mathematical Yang-Lee theorem, physical microscopic explanation of the liquid to gas phase transition sheds sharp light on the physical origin of the transition, the sudden change in the inhomogeneous population of the 3Ndimensional configuration space.

The liquid-solid phase transition.
In contrast to the liquid phase, in the crystal phase a molecule can only move locally within its lattice cage of the size d 3 instead of the whole volume V 0N of the condensate. I.e. in equation (26)

Summary of section IV.
The gas-liquid transition is linked to the transition from uniform filling of the container volume, V, by the gas to the smaller eigen-volume of the system, V 0 , in its condensed phase when the system is inhomogeneous (some liquid drops inside the larger empty volume V). At the surface of a drop φ int > E 0 = min[φ int ], i.e. the surface gives a negative contribution to S int in equation (22) and to S at energies E ≥ E 0 , as was similarly assumed in section 3 and explicitly in equation (15).

Acknowledgement.
D. H. E. Gross is grateful to J. Möller for insistent, therefore helpful, discussions.

Appendix A. A simple model for condensation.
Assume the various potential-pockets are attractive square-wells like with depths Φ λ < 0.
All particles condensed into single clusters λ = 1 will have the lowest potential Φ 1 and consequently the largest E -Φ. On the other hand the volume factor ~V/V 0(N-1) appears only once.
In terms of I(K), the partition function W(E) and its derivatives are expressed as: To get a convexity (β ' ≥ 0), we must have: ( ) ( ) ( ) When calculating equations (32)-(34), one immediately sees that for a sharply monodispersed sum (28), λ ~ λ 0 (single phase): ( ) ( ) ( ) equal to the ideal gas value, independently of whether the system is condensed or not. (Remember, in equation (28) we assumed for simplicity that Φ λ has a flat bottom).