Nonequilibrium Thermodynamics. Symmetric and Unique Formulation of the First Law, Statistical Definition of Heat and Work, Adiabatic Theorem and the Fate of the Clausius Inequality: A Microscopic View

Abstract

The status of heat and work in nonequilibrium thermodynamics is quite confusing and nonunique at present with conflicting interpretations even after a long history of the first law dE(t) = d e Q(t) − dW e (t) in terms of exchange heat and work, and is far from settled. Moreover, the exchange quantities lack certain symmetry (see text). By generalizing the traditional concept to also include their time-dependent irreversible components d i Q(t) and d i W (t) allows us to express the first law in a symmetric form dE(t) = dQ(t) − dW (t) in which dQ(t) and work dW (t) appear on equal footing and possess the symmetry. We prove that d i Q(t) ≡ d i W (t); as a consequence, irreversible work turns into irreversible heat. Statistical analysis in terms of microstate probabilities p i (t) uniquely identifies dW (t) as isentropic and dQ(t) as isometric (see text) change in dE(t), a result known in equilibrium. We show that such a clear separation does not occur for d e Q(t)and dW e (t). Hence, our new formulation of the first law provides tremendous advantages and results in an extremely useful formulation of non-equilibrium thermodynamics, as we have shown recently [Phys. Rev. E 81, 051130 (2010); ibid 85, 041128 and 041129 ( 2012)]. We prove that an adiabatic process does not alter p i . All these results remain valid no matter how far the system is out of equilibrium. When the system is in internal equilibrium, dQ(t) ≡ T (t)dS(t) in terms of the instantaneous temperature T (t) of the system, which is reminiscent of equilibrium, even though, neither d e Q(t) ≡ T (t)d e S(t) nor d i Q(t) ≡ T (t)d i S(t). Indeed, d i Q(t) and d i S(t) have very different physics. We express these quantities in terms of d e p i (t) and d i p i (t), and demonstrate that p i (t) has a form very different from that in equilibrium. The first and second laws are no longer independent so that we need only one law, which is again reminiscent of equilibrium. The traditional formulas like the Clausius inequality d e Q(t)/T 0 < 0, ∆ e W < −∆ [E(t − T 0 S(t))], etc. become equalities dQ(t)/T (t) ≡ 0, ∆W = −∆ [E(t − T (t)S(t)], etc, a quite remarkable but unexpected result in view of the fact that ∆ i S(t) > 0. We identify the uncompensated transformation N (t, τ ) during a cycle. We determine the irreversible components in two simple cases to show the usefulness of our approach; here, the traditional formulation is of no use. Our extension bring about a very strong parallel between equilibrium and non-equilibrium thermodynamics, except that one has irreversible entropy generation d i S(t) > 0 in the latter.

First-principles nonequilibrium deterministic equation of motion of a Brownian particle and microscopic viscous drag

“…To obtain the µNEQT, we need to uniquely extract from the MNEQT a description suitable at the microstate level. We have introduced the µNEQT a while back [8,39,50,51]. It is a first-principles theory and its main purpose here is to study BP in NEQ situations, where the Langevin equation in Eqs.…”

A Novel Trick to Overcome the Phase Space Volume Change and the Use of Hamiltonian Trajectories with an emphasis on the Free Expansion

“…In conclusion, it is not a surprise that we are left to exclusively use the MNEQT and µNEQT in this study of interacting and isolated systems. We have already applied the MNEQT to briefly study free expansion [42,43]. Here, we wish to go beyond the earlier study to demonstrate how the µNEQT can be used to study expansion/contraction with special attention to free expansion by including internal variables also.…”

“…This is in a NEQ macrostate until ξ achieves its EQ value ξ = 0 during P, at the end of which at t = τ eq the gas eventually comes into M fin,eq isoenergetically. We briefly review this expansion in the MNEQT [43].…”

Hybrid Einstein-Langevin Approach for Microscopic formulation of Viscous Drag: An Alternative to the Langevin Equation

“…To summarize, we have given an alternative to the Langevin equation based on µNEQT that was initiated a while back [26,29,31]. Its usage shows that the uniquely defined microforces and microworks in the system are, as expected, fluctuating quantities.…”

“…We will treat the piston problem for simplicity as it is commonly discussed in introductory physics. Let x denote a phase point in the phase space of Σ so that the Hamiltonian of the system is written as H( x| V, P BP , P R ) in which V, P BP and P R appear as parameters: the variations dV, dP BP and dP R change the value of H; this change represents the generalized work dW done by the system as shown elsewhere [25][26][27][28][29][30][31]. Introducing the system-intrinsic (SI) "mechanical forces" obtained directly from the SI-Hamiltonian…”