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arXiv:0711.4176 · Physical Review E

Unified approach to the derivation of work theorems for equilibrium and steady-state, classical and quantum Hamiltonian systems

Maxim F. Gelin, Daniel S. Kosov

Published Jul 18, 2008• 24 citations
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Abstract

We present a unified and simple method for deriving work theorems for classical and quantum Hamiltonian systems, both under equilibrium conditions and in a steady state. Throughout the paper, we adopt the partitioning of the total Hamiltonian into the system part, the bath part, and their coupling. We rederive many equalities which are available in the literature and obtain a number of new equalities for nonequilibrium classical and quantum systems. Our results can be useful for determining partition functions and (generalized) free energies through simulations or measurements performed on nonequilibrium systems.

DOI
10.1103/physreve.78.011116

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We present a unified and simple method for deriving work theorems for classical and quantum Hamiltonian systems, both under equilibrium conditions and in a steady state. Throughout the paper, we adopt the partitioning of the total Hamiltonian into the system part, the bath part, and their coupling. We rederive many equalities which are available in the literature and obtain a number of new equalities for nonequilibrium classical and quantum systems. Our results can be useful for determining partition functions and (generalized) free energies through simulations or measurements performed on nonequilibrium systems.
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We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
The first part of this thesis focuses on verifying the quantum nonequilibrium work relation in the presence of decoherence. The nonequilibrium work relation is a generalization of the second law of thermodynamics that links nonequilibrium work measurements to equilibrium free energies via an equality. Despite being well established for classical systems, a quantum work relation is conceptually difficult to construct for systems that interact with their environment. We argue that for a quantum system which undergoes decoherence but not dissipation, these conceptual difficulties do not arise and the work relation can be proven similarly to the case of an isolated system. This result is accompanied by an experimental demonstration using trapped ions. The second part of this thesis examines the relationship between quantum work and coherence by constructing analogous quantities in classical physics. It has recently been shown that quantum coherence can function as a resource for work extraction. Furthermore, it has been suggested that this property could be a truly quantum aspect of thermodynamics with no classical analog. We examine this assertion within the framework of classical Hamiltonian mechanics and canonical quantization. For classical states we define a so called non-uniformity measure and show that it is a resource for work extraction similar to quantum coherence. Additionally, we show that work extracted from non-uniformity and coherence agree in the classical limit. This calls into question the idea that coherence qualitatively separates classical and quantum thermodynamics. The final part of this thesis explores the connection between decoherence and adiabatic (quasistatic) driving. This topic is inspired by an experiment where it was seen that strong dephasing suppressed energy level transitions. Using a perturbative method we investigate this mechanism in the regime of small to moderate decoherence rate and ask if decoherence can help suppress energy transitions when compared with an adiabatic process without decoherence. We find that strategies that include decoherence are inferior to those where decoherence is absent.
The dynamics of a quantum system coupled to a classical environment and subject to constraints that drive it out of equilibrium are described. The evolution of the system is governed by the quantum-classical Liouville equation. Rather than evaluating the evolution of the mixed quantum-classical density operator, we derive exact equations of motion for the nonequilibrium average values of a set of operators or variables, along with correlation function expressions for the dissipative coefficients that enter these equations. These equations are obtained by requiring that the exact nonequilibrium averages are equal to local nonequilibrium averages that depend on auxiliary fields whose values satisfy evolution equations obtained using projection operator methods. The results are illustrated by deriving reaction-diffusion equations coupled to fluid hydrodynamic equations for a solution of quantum particles that can exist in two metastable states. Nonequilibrium steady states are discussed along with the reaction rate and diffusion correlation functions that characterize such states.