In this paper, we take a control-theoretic approach to answering some standard questions in statistical mechanics, and use the results to derive limitations of classical measurements. A central problem is the relation between systems which appear macroscopically dissipative but are microscopically lossless. We show that a linear system is dissipative if, and only if, it can be approximated by a linear lossless system over arbitrarily long time intervals. Hence lossless systems are in this sense dense in dissipative systems. A linear active system can be approximated by a nonlinear lossless system that is charged with initial energy. As a by-product, we obtain mechanisms explaining the Onsager relations from time-reversible lossless approximations, and the fluctuation-dissipation theorem from uncertainty in the initial state of the lossless system. The results are applied to measurement devices and are used to quantify limits on the so-called observer effect, also called back action, which is the impact the measurement device has on the observed system. In particular, it is shown that deterministic back action can be compensated by using active elements, whereas stochastic back action is unavoidable and depends on the temperature of the measurement device.
@cite_13: Exact generalized Langevin equations are derived for arbitrarily nonlinear systems interacting with specially chosen heat baths. An example is displayed in which the Langevin equation is nonlinear but approximately Markovian.
@cite_35: The quantum Langevin equation is the Heisenberg equation of motion for the (operator) coordinate of a Brownian particle coupled to a heat bath. We give an elementary derivation of this equation for a simple coupled-oscillator model of the heat bath.
@cite_37: 
@cite_26: Thermodynamics is universal;in principle, the subject appliesto everything. Accordingly,everyone is entitled to theirown opinion regarding howthermodynamics relates tohis her own field of specializa-tion. According to Albert Einstein “thermodynamics is theonly physical theory of a universal nature of which I amconvinced that it will never be overthrown.” Throughout history, universal principles such as basichuman rights have not always been implemented in prac-tice. Thermodynamics, however, is independent of one’spolitical philosophy. No tyrant or dictator can decide aboutthermodynamics. If someone comes to me with a rheologi-cal law for a non-Newtonian fluid that does not obey thesecond law of thermodynamics, I can say beforehand thatsomething is fishy with that model. What is left to humans,and especially to the scientist, is the explanation of the emer-gence of thermodynamics in its irreversible version from anapparently nondissipative microworld. (Recall that the lawsof physics are time reversible at the microscale.) Moreover,the job of the scientist also entails developing and interpret-ing the most appropriate mathematical framework for ther-modynamics as well as applying its central tenets todisputed fields such as living systems and large-scalephysics, that is, cosmology and the future evolution of theuniverse—and, therefore, the ultimate destiny of humans. It is an agreed fact that mathematics is the common lan-guage of science. The authors claim that “mathematics isthe irrefutable pathway to the truth.” This statement is notexactly true in general; it is true only insofar as mathemat-ics
@cite_18: 
@cite_33: We apply the influence-functional method of Feynman and Vernon to the study of Brownian motion at arbitrary temperature. By choosing a specific model for the dissipative interaction of the system of interest with its environment, we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient. We show that in the limit h→0 the results obtained from the influence functional formalism reduce to the classical Fokker-Planck equation. In the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature, we obtain an explicit expression for the time evolution of the complete density matrix ϱ(x, x′, t) when the system starts in a particular kind of pure state. We compare our results with those of other approaches to the problem of dissipation in quantum mechanics.
@cite_38: The purpose of this paper is to highlight the central role that the time asymmetry of stability plays in feedback control. We show that this provides a new perspective on the use of doubly-infinite or semi-infinite time axes for signal spaces in control theory. We then focus on the implication of this time asymmetry in modeling uncertainty, regulation and robust control. We point out that modeling uncertainty and the ease of control depend critically on the direction of time. We also discuss the relationship of this control-based time-arrow with the well known arrows of time in physics.
@cite_8: 
@cite_28: The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper.
@cite_42: In this paper, we take a control-theoretic approach to answering some standard questions in statistical mechanics. A central problem is the relation between systems which appear macroscopically dissipative but are microscopically lossless. We show that a linear macroscopic system is dissipative if and only if it can be approximated by a linear lossless microscopic system, over arbitrarily long time intervals. As a by-product, we obtain mechanisms explaining Johnson-Nyquist noise as initial uncertainty in the lossless state, as well as measurement back action and a trade-off between process and measurement noise.
@cite_40: The principle of equipartition of energy is usually viewed as a statistical result formulated in terms of the probability distribution of configurations, that is, entropy. In this paper we use deterministic linear systems techniques to analyze the vibrational energy of systems of undamped coupled oscillators with identical coupling. Our approach is based on time averaging of squared outputs of the system and thus avoids both recurrence and statistical arguments. We first consider a single undamped oscillator and show that the time averaged potential energy and the time-averaged kinetic energy converge to the same value. Next, we consider a collection of n identical undamped oscillators with lossless coupling. As in the case of a single oscillator, equipartition of energy holds for the total system kinetic and potential energies. We focus on the equipartition of oscillator energy, that is, the equal distribution of energy among oscillators, regardless of the form of energy. We derive expressions for the transient and steady-state behavior of each oscillator.
@cite_23: This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.
@cite_2: An apparent paradox in classical statistical physics is the mechanism by which conservative, time-reversible microscopic dynamics, can give rise to seemingly dissipative behavior. In this paper we use system theoretic tools to show that dissipation can arise as an artifact of incomplete observations over a finite horizon. In addition, this approach allows us to obtain finite-time, low order, approximations of systems with moderate size and to establish how the approach to the thermodynamic limit depends on the different physical parameters.
@cite_10: The second law of thermodynamics is studied from the point of view of stochastic control theory. We find that the feedback control laws which are of interest are those which depend only on average values, and not on sample path behavior. We are lead to a criterion which, when satisfied, permits one to assign a temperature to a stochastic system in such a way as to have Carnot cycles be the optimal trajectories of optimal control problems. Entropy is also defined and we are able to prove an equipartition of energy theorem using this definition of temperature. Our formulation allows one to treat irreversibility in a quite natural and completely precise way.
@cite_17: We rigorously derive the main results of thermo-dynamics, including Carnot's theorem, in the framework of time-varying linear systems.
Substantial work has already been done in the control community in formulating various results of classical thermodynamics in a more mathematical framework. In @cite_10 @cite_8 , the second law of thermodynamics is derived and a control-theoretic heat engine is obtained (in @cite_17 these results are generalized). In @cite_26 , a rigorous dynamical systems approach is taken to derive the laws of thermodynamics using the framework of dissipative systems @cite_28 @cite_23 . In @cite_37 , it is shown how the entropy flows in Kalman-Bucy filters, and in @cite_18 Linear-Quadratic-Gaussian control theory is used to construct heat engines. In @cite_40 @cite_2 @cite_42 , the problem of how lossless systems can appear dissipative (compare with @cite_13 @cite_35 @cite_33 above) is discussed using various perspectives. In @cite_38 , how the direction of time affects the difficulty of controlling a process is discussed.