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Ensemble and Trajectory Thermodynamics: a Brief Introduction C

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Ensemble and Trajectory Thermodynamics: a Brief Introduction C Physica A 418 (2015) 6–16 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Ensemble and trajectory thermodynamics: A brief introduction C. Van den Broeck a,∗, M. Esposito b a Hasselt University, B-3590 Diepenbeek, Belgium b Complex Systems and Statistical Mechanics, University of Luxembourg, L-1511 Luxembourg, Luxembourg h i g h l i g h t s • We review stochastic thermodynamics at the ensemble level. • We formulate first and second laws at the trajectory level. • The stochastic entropy production is the log-ratio of trajectory probabilities. • We establish the relation between the stochastic grand potential, work and entropy production. • We derive detailed and integral fluctuation theorems. article info a b s t r a c t Article history: We revisit stochastic thermodynamics for a system with discrete energy states in contact Available online 24 April 2014 with a heat and particle reservoir. ' 2014 Elsevier B.V. All rights reserved. Keywords: Stochastic Thermodynamics Entropy Fluctuation theorem Markov process 1. Introduction Over the last few years, it has become clear that one can extend thermodynamics, which is traditionally confined to the description of equilibrium states of macroscopic systems or to the transition between such states, to cover the nonequilib- rium dynamics of small scale systems. This extension has been carried out at several levels of description including systems described by discrete and continuous Markovian and non-Markovian stochastic dynamics, by classical Hamiltonian and quantum Hamiltonian dynamics and by thermostatted systems. These developments can be seen, on one side, as extend- ing the work of Onsager and Prigogine [1–4] by including microscopic dynamical properties into the far from equilibrium irreversible realm. On the other side, they have led to the reassessment of the cornerstone of thermodynamics, namely the second law of thermodynamics, which is replaced by a much deeper symmetry relation, embodied in the integral and de- tailed fluctuation theorems. On the more practical side, the new formulation allows to address new questions, either related to nonequilibrium properties, such as efficiency at maximum power or information to work conversion, or relating to the thermodynamic description of small systems, e.g., the discussion of Brownian motors and refrigerators or the efficiency of quantum dots and other small scale devices. In this paper, we present a brief introduction to stochastic thermodynamics. We refer to the literature for more advanced reviews [5–13]. ∗ Corresponding author. Tel.: +32 11268214. E-mail address: [email protected] (C. Van den Broeck). http://dx.doi.org/10.1016/j.physa.2014.04.035 0378-4371/' 2014 Elsevier B.V. All rights reserved. C. Van den Broeck, M. Esposito / Physica A 418 (2015) 6–16 7 2. Ensemble thermodynamics We consider a system, with discrete non-degenerate states, in contact with a single (ideal, non-dissipative) heat and particle reservoir at temperature T (β D 1=.kBT /) and chemical potential µ. The states are identified by an index m, with corresponding energy ϵm and particle number nm. For simplicity we consider a single type of particle. We assume that the system can also exchange work with an (ideal, non-dissipative) work source which controls its energy levels ϵ(λ) via a time-dependent control parameter λ D λ(t/. The particle number in a given state is however supposed to be fixed. In the ensemble picture, the state of the system is described by a probability distribution Pm to be in the state m, with P D m Pm 1. Note that this distribution does not have to be of the equilibrium form, so that ``traditional equilibrium con- cepts'', such as temperature and chemical potential need not exist for the system. They are however well defined for the ideal reservoir, and when appearing in the formulas below, T and µ refer to this reservoir. The time evolution of the state is described by a Markovian master equation: X dt Pm D Wm;m0 Pm0 : (1) m0 0 Here Wm;m0 is the probability per unit time to make a transition from state m to m. We use the shorthand notation with D − P 0 diagonal elements defined as Wm;m m06Dm Wm ;m. Alternatively: X Wm;m0 D 0; (2) m a property that guarantees the conservation of normalization. The transition rates have to satisfy an additional property. In the steady state, the system is at equilibrium with the reservoir. Statistical physics prescribes that the steady state distribu- eq tion is given by the grand canonical equilibrium distribution Pm [14]: eq D {− − − eq g Pm exp β(ϵm µnm Ω / : (3) The (equilibrium) grand potential Ωeq follows from the normalization of P eq: eq X exp{−βΩ g D exp{−β(ϵm − µnm/g: (4) m The crucial property that is required from the rates is the so-called condition of detailed balance, i.e., at equilibrium every transition, say from m to m0, and its inverse, from m0 to m, have to be equally likely: eq 0 D 0 eq Wm;m Pm0 Wm ;mPm : (5) Combined with the explicit expression of the equilibrium distribution, this gives: Wm0;m ϵm − ϵm0 − µ(nm − nm0 / qm;m0 kB ln D D : (6) Wm;m0 T T 0 qm;m0 is the ``elementary'' heat absorbed by the system to make the transition from m to m. We stress that in the presence of driving, which is shifting the energy levels in time, this relation is supposed to hold at each moment in time, hence the rates also become time-dependent. This condition will be crucial to obtain the correct formulation of the second law. We next introduce the basic state functions – quantities that depend on probability distribution Pm of the system, but not on the way this distribution was achieved – namely the ensemble-averaged and in general nonequilibrium values of energy, particle number and entropy: X E D ϵmPm D hϵmi; (7) m X N D nmPm D hnmi; (8) m X S D −kB Pm ln Pm D ⟨−kB ln Pmi: (9) m It is clear from the above formulas that these state variables can change due to two different mechanisms: a change in occupation of the levels, i.e., a modification of Pm, or a shift of the energy levels, i.e., a change of the energy level ϵm. In particular, the rate of energy change is given by: X dt E D fϵmdt Pm C dt ϵmPmg (10) m P P P D Q C Wchem C W : (11) 8 C. Van den Broeck, M. Esposito / Physica A 418 (2015) 6–16 P P This decomposition reproduces the first law. Work rate (power) W , particle flow dt N and heat flow Q are given by: P X W D dt ϵmPm D dt λ dλE; (12) m X dt N D nmdt Pm; (13) m P X P Q D ϵmdt Pm − Wchem; (14) m P with the chemical work rate Wchem P Wchem D µdt N: (15) In words, work is the result of the energy shift of an occupied state. Heat and chemical work correspond to transitions between states, implying a change in occupation probability. As is well known, heat and work are not state functions (or the difference of state functions) as they depend on the way the transition between states is carried out. We also mention for later use the following expression for the heat flux, cf. (13)–(15): P X Q D (ϵm − µnm/dt Pm: (16) m Turning to the second law, we will show that the above definition of nonequilibrium entropy is a proper choice that reduces to the standard thermodynamic entropy at equilibrium, but with the additional advantage that it preserves – in nonequilibrium – the basic features of the second law, namely the relation between heat and entropy exchange and the positivity of the entropy production. Explicitly: P P dt S D Se C Si; (17) P P Se D Q =T ; (18) P Si ≥ 0: (19) The proof goes as follows: X X dt S D −kB dt Pm ln Pm D −kB Wm;m0 Pm0 ln Pm m m;m0 X Pm D −kB Wm;m0 Pm0 ln P 0 m;m0 m kB X Pm0 D Wm;m0 Pm0 − Wm0;mPm ln 2 P m;m0 m kB X Wm;m0 Pm0 kB X Wm0;m D Wm;m0 Pm0 − Wm0;mPm ln C Wm;m0 Pm0 − Wm0;mPm ln : (20) 2 W 0 P 2 W 0 m;m0 m ;m m m;m0 m;m We thus identify the entropy production and entropy flow as: P kB X Wm;m0 Pm0 Si D Wm;m0 Pm0 − Wm0;mPm ln ; (21) 2 W 0 P m;m0 m ;m m P kB X Wm0;m Se D Wm;m0 Pm0 − Wm0;mPm ln ; (22) 2 W 0 m;m0 m;m or: P X Si D Jm;m0 Xm;m0 ; (23) m>m0 P X qm;m0 Se D Jm;m0 ; (24) T m>m0 with Jm;m0 D Wm;m0 Pm0 − Wm0;mPm; (25) Wm;m0 Pm0 Xm;m0 D kB ln ; (26) Wm0;mPm qm;m0 D ϵm − ϵm0 − µ(nm − nm0 /: (27) C. Van den Broeck, M. Esposito / Physica A 418 (2015) 6–16 9 0 P Jm;m0 is the net rate of transitions from m to m and Xm;m0 the corresponding thermodynamic force. Si is clearly non-negative. In fact, a stronger property holds: it is the sum of contributions due to all pairwise transitions between different states m 0 P and m , and each of these terms is non-negative. Concerning Se, one finds using (6) and (16): P X Wm0;m Se D kB Wm;m0 Pm0 ln (28) W 0 m;m0 m;m X ϵm − ϵm0 − µ(nm − nm0 / D Wm;m0 Pm0 (29) T m;m0 X ϵm − µnm D Wm;m0 Pm0 (30) T m;m0 P X ϵm − µnm Q D dt Pm D ; (31) m T T which is indeed the expected expression for the entropy flow.
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