We consider a time-dependent quantum linear oscillator coupled to a bath at an arbitrary strength. We then introduce a generalized Jarzynski equality (GJE) which includes the terms reflecting the system-bath coupling. This enables us to study systematically the coupling effect on the linear oscillator in a non-equilibrium process. This is also associated with the second law of thermodynamics beyond the weak-coupling limit. We next take into consideration the GJE in the classical limit. By this generalization we show that the Jarzynski equality in its original form can be associated with the second law, in both quantal and classical domains, only in the vanishingly small coupling regime.
We consider a time-dependent quantum linear oscillator coupled to a bath at an arbitrary strength. We then introduce a generalized Jarzynski equality (GJE) which includes the terms reflecting the system-bath coupling. This enables us to study systematically the coupling effect on the linear oscillator in a non-equilibrium process. This is also associated with the second law of thermodynamics beyond the weak-coupling limit. We next take into consideration the GJE in the classical limit. By this generalization we show that the Jarzynski equality in its original form can be associated with the second law, in both quantal and classical domains, only in the vanishingly small coupling regime.
We consider a quantum linear oscillator coupled at an arbitrary strength to a bath at an arbitrary temperature. We find an exact closed expression for the oscillator density operator. This state is noncanonical but can be shown to be equivalent to that of an uncoupled linear oscillator at an effective temperature T*(eff) with an effective mass and an effective spring constant. We derive an effective Clausius inequality deltaQ*(eff)< or =T*(eff)dS , where deltaQ*(eff) is the heat exchanged between the effective (weakly coupled) oscillator and the bath, and S represents a thermal entropy of the effective oscillator, being identical to the von-Neumann entropy of the coupled oscillator. Using this inequality (for a cyclic process in terms of a variation of the coupling strength) we confirm the validity of the second law. For a fixed coupling strength this inequality can also be tested for a process in terms of a variation of either the oscillator mass or its spring constant. Then it is never violated. The properly defined Clausius inequality is thus more robust than assumed previously.
This paper revisits the classical problem of representing a thermal bath interacting with a system as a large collection of harmonic oscillators initially in thermal equilibrium. As is well known, the system then obeys an equation, which in the bulk and in the suitable limit tends to the Kramers–Langevin equation of physical kinetics. I consider time-dependent system-bath coupling and show that this leads to an additional harmonic force acting on the system. When the coupling is switched on and switched off rapidly, the force has delta-function support at the initial and final time. I further show that the work and heat functionals as recently defined in stochastic thermodynamics at strong coupling contain additional terms depending on the time derivative of the system-bath coupling. I discuss these terms and show that while they can be very large if the system-bath coupling changes quickly, they only give a finite contribution to the work that enters in Jarzynski’s equality. I also discuss that these corrections to standard work and heat functionals provide an explanation for non-standard terms in the change of the von Neumann entropy of a quantum bath interacting with a quantum system found in an earlier contribution (Aurell and Eichhorn, 2015).
The nonequilibrium fluctuation relation is a cornerstone of quantum thermodynamics. It is widely believed that the system-bath heat exchange obeys the famous Jarzynski-Wójcik fluctuation theorem. However, this theorem is established in the Born-Markovian approximation under the weak-coupling condition. Via studying the energy exchange between a harmonic oscillator and its coupled bath in the non-Markovian dynamics, we establish a generalized quantum fluctuation theorem for energy exchange being valid for arbitrary coupling strength. The Jarzynski-Wójcik fluctuation theorem is recovered in the weak-coupling limit. We also find the average energy exchange exhibits rich nonequilibrium characteristics when different numbers of system-bath bound states are formed, which suggests a useful way to control the quantum heat. Deepening our understanding of the fluctuation relation in quantum thermodynamics, our result lays the foundation to design high-efficiency quantum heat engines.
In this work, we study the fluctuation relation and the second law of thermodynamics within a quantum linear oscillator externally driven over the period of time t = tau. To go beyond the standard approach (the two-point projective measurement one) to this subject and also render it discussed in both quantum and classical domains on the single footing, we recast this standard approach in terms of the Wigner function and its propagator in the phase space (x,p). With the help of the canonical transformation from (x,p) to the angle-action coordinates (ϕ,I), we can then derive a measurement-free (classical-like) form of the Crooks fluctuation relation in the Wigner representation. This enables us to introduce the work W_{I_0,I_{tau}} associated with a single run from (I_0) to (I_{tau}) over the period tau, which is a quantum generalization of the thermodynamic work with its roots in the classical thermodynamics. This quantum work differs from the energy difference e_{I_0,I_{tau}} = e(I_{tau}) - e(I_0) unless beta, hbar --> 0. Consequently, we will obtain the quantum second-law inequality Delta F_{beta} \leq _{P} \leq _{P} = Delta U, where P, Delta F_{beta}, and _P denote the work (quasi)-probability distribution, the free energy difference, and the average work distinguished from the internal energy difference Delta U, respectively, while _P --> Delta U in the limit of beta, hbar --> 0 only. Therefore, we can also introduce the quantum heat Q_q = Delta U - W even for a thermally isolated system, resulting from the quantum fluctuation therein. This is a more fine-grained result than _P = Delta U obtained from the standard approach. Owing to the measurement-free nature of the thermodynamic work W_{I_0,I_{tau}}, our result can also apply to the (non-thermal) initial states rho_0 = (1-gamma) rho_{beta} + gamma sigma with sigma \ne rho_{beta}.
Quantum systems strongly coupled to many-body systems equilibrate to the reduced state of a global thermal state, deviating from the local thermal state of the system as it occurs in the weak-coupling limit. Taking this insight as a starting point, we study the thermodynamics of systems strongly coupled to thermal baths. First, we provide strong-coupling corrections to the second law applicable to general systems in three of its different readings: As a statement of maximal extractable work, on heat dissipation, and bound to the Carnot efficiency. These corrections become relevant for small quantum systems and vanish in first order in the interaction strength. We then move to the question of power of heat engines, obtaining a bound on the power enhancement due to strong coupling. Our results are exemplified on the paradigmatic non-Markovian quantum Brownian motion.
Thermodynamics is the phenomenological theory of heat and work. Here we analyze to what extent quantum thermodynamic relations are immune to the underlying mathematical formulation of quantum mechanics. As a main result, we show that the Jarzynski equality holds true for all non-hermitian quantum systems with real spectrum. This equality expresses the second law of thermodynamics for isothermal processes arbitrarily far from equilibrium. In the quasistatic limit however, the second law leads to the Carnot bound which is fulfilled even if some eigenenergies are complex provided they appear in conjugate pairs. Furthermore, we propose two setups to test our predictions, namely with strongly interacting excitons and photons in a semiconductor microcavity and in the non-hermitian tight-binding model.