Microscopic analysis of the Clausius-Duhem inequality

Given a general thermodynamic process which carries a system from one equilibrium state to another, we construct a quantity whose average, over an ensemble of microscopic realizations of the process, depends only on these end states, even if at intermediate times the system is out of equilibrium. This result leads directly to a statistical statement of the Clausius-Duhem inequality, and can be generalized to situations in which the system begins and/or ends in nonequilibrium states. 05.70.Ln, 05.20.-y, 82.60.-s Typeset using REVTEX 1 The Clausius-Duhem inequality of classical thermodynamics – a statement of the Second Law – applies to any thermodynamic process which carries a system from one equilibrium state (A) to another (B). It states that the integrated heat absorbed by the system, inversely weighted by the temperature at which that heat is absorbed, is bounded from above by the net change in the entropy of the system: ∫ B A dQ T ≤ ∆S ≡ S − S. (1) By “thermodynamic process”, we have in mind a situation in which the system is brought into thermal contact with a sequence of heat reservoirs at different temperatures, one at a time, while one or more external parameters of the system are varied with time; the denominator in Eq.1 then denotes the temperature of the reservoir from which the system absorbs a quantity of heat dQ. In general, the process is carried out over a finite time, with a finite number of reservoirs, and it is not assumed that the system is given sufficient time to reach thermal equilibrium with each reservoir. The system thus generally evolves through a sequence of non-equilibrium states during the thermodynamic process. The aim of the present paper is a microscopic analysis of the Clausius-Duhem (CD) inequality, explicitly taking into account all degrees of freedom involved. This analysis will require a statistical approach: we will work with an ensemble of microscopic realizations of the thermodynamic process. Each realization is described by a trajectory, completely specifying the evolution of all the degrees of freedom which make up the system of interest and participating reservoirs. We will consider the ensemble of such realizations consistent with the macroscopic preparation of the system and reservoirs. Each realization thus represents a possible “microscopic history”, given that we prepare the system and reservoirs, and execute the thermodynamic process, according to a particular set of macroscopic instructions. A statistical ensemble of realizations implies fluctuations – from one realization to another – of various quantities of physical interest. For instance, the exact value of the integral ∫B A dQ/T will differ, for different microscopic realizations of the same thermodynamic process. For macroscopic systems, it is often true that the ensemble average of some quantity of physical interest is the experimentally observed value, and the differences in its precise value – from one realization to the next – represent tiny fluctuations around that average. In the context of the CD inequality, this suggests that, for a thermodynamic process beginning and ending in equilibrium states, the average of the integral ∫B A dQ/T (over microscopic realizations of the process) will be bounded from above by the net change in the thermodynamic entropy: 〈 ∫B A dQ/T 〉 ≤ ∆S, where angular brackets denote the ensemble average. We will show that, in fact, this result follows directly from a straightforward microscopic analysis of thermodynamic processes. Indeed, we will show that this inequality is the immediate consequence of a stronger result, Eq.9 below, which is really the central result of this paper. As we will further discuss, Eq.9 also implies that the probability of observing a violation of the CD inequality, by an amount greater than Γ0, is bounded from above by e −Γ0/kB (where kB is the Boltmann constant), so that macroscopic violations are exceedingly rare. We will then generalize this analysis to the situation in which the preparation and execution of the thermodynamic process are such that the system of interest does not necessarily begin and end in states of equilibrium; in that case we will show that 〈 ∫ dQ/T 〉 is no greater than the net change in the statistical entropy (− ∫ ρ ln ρ) of the phase space distributions describing the initial and final statistical states of the system. 2 We will work with the following picture. We imagine N finite objects which will play the role of heat reservoirs in our analysis. We will refer to these as “bricks” which have been prepared at finite temperatures Tn, n = 1, 2, · · · , N . By this we mean that, prior to the execution of the thermodynamic process, each brick is allowed to come to thermal equilibrium with an infinite heat reservoir, then isolated. We make no assumptions about the set of temperatures, T1, T2, · · · , TN , at which the N bricks are so prepared. In keeping with our view of the bricks as heat “reservoirs”, we assume each brick has a heat capacity far greater than that of the system of interest, although this assumption is not really necessary for the derivation of our central result. For purpose of illustration, we will take our system of interest to be a gas of interacting particles inside a container, one end of which is closed off by a movable piston. The position of the piston will be treated as an external parameter, denoted by λ. We prepare the gas by allowing it to come to thermal equilibrium with an infinite reservoir at a temperature T, while holding the piston fixed at a position λ. At the macroscopic level, thus, the gas begins in an equilibrium state, specified by the pair of values (λ, T). The thermodynamic process then consists of a sequence of steps during which the container with gas is placed in contact with the bricks, one at a time, while the position of the piston is externally varied along a pre-determined path λ(t), over a time interval [0, τ ∗]. Thus, λ(0) = λ, and λ(τ ∗) = λ, where λ is the final value of the external parameter. Let n(t) identify the brick with which the system is in thermal contact at time t; assume that n(t) increases in unit steps from n(0) = 1 to n(τ ∗) = N . Once the piston has reached its final position (λ), we hold it fixed, and allow the gas to reach thermal equilibrium with the Nth brick. The thermodynamic process then ends once equilibrium has been achieved, at a time τ > τ ∗. The gas thus ends in an equilibrium state (λ, T), where T = TN . To analyze this process at the microscopic level, let z denote a point in the phase space of the gas – specifying the positions and momenta of all the gas particles – and let zn denote a point in the phase space of the nth brick. Let y = (z, z1, · · · , zN) then specify a point in the full phase space, describing the instantaneous state of all degrees of freedom involved. Evolution in the full phase space is governed by a time-dependent Hamiltonian H(y, t), which we may assume to take the form: H(y, t) = Hλ(t)(z) + N