Increase of Boltzmann entropy in a quantum forced harmonic oscillator

Evidence for simple arrow of time functions in closed chaotic quantum systems

Through an explicit construction, we assign to any infinite temperature autocorrelation function C(t) a set of functions α^{n}(t). The construction of α^{n}(t) from C(t) requires the first 2n temporal derivatives of C(t) at times 0 and t. Our focus is on α^{n}(t) that (almost) monotonously decrease; we call these "arrows of time functions" (AOTFs). For autocorrelation functions of few-body observables we numerically observe the following: An AOTF featuring a low n may always be found unless the system is in or close to a nonchaotic regime with respect to a variation of some system parameter. All α^{n}(t) put upper bounds to the respective autocorrelation functions, i.e., α^{n}(t)≥C^{2}(t). Thus the implication of the existence of an AOTF is comparable to that of the H theorem, as it indicates a directed approach to equilibrium. We furthermore argue that our numerical finding may to some extent be traced back to the operator growth hypothesis. This argument is laid out in the framework of the so-called recursion method.

Computing characteristic functions of quantum work in phase space

In phase space, we analytically obtain the characteristic functions (CFs) of a forced harmonic oscillator [Talkner et al., Phys. Rev. E 75, 050102(R) (2007)PLEEE81539-375510.1103/PhysRevE.75.050102], a time-dependent mass and frequency harmonic oscillator [Deffner and Lutz, Phys. Rev. E 77, 021128 (2008)PLEEE81539-375510.1103/PhysRevE.77.021128], and coupled harmonic oscillators under driving forces in a simple and unified way. For general quantum systems, a numerical method that approximates the CFs to ℏ^{2} order is proposed. We exemplify the method with a time-dependent frequency harmonic oscillator and a family of quantum systems with time-dependent even power-law potentials.

Stiffness of probability distributions of work and Jarzynski relation for non-Gibbsian initial states

We consider closed quantum systems (into which baths may be integrated) that are driven, i.e., subject to time-dependent Hamiltonians. Our point of departure is the assumption that if systems start in non-Gibbsian states at some initial energies, the resulting probability distributions of work may be largely independent of the specific initial energies. It is demonstrated that this assumption has some far-reaching consequences, e.g., it implies the validity of the Jarzynski relation for a large class of non-Gibbsian initial states. By performing numerical analysis on integrable and nonintegrable spin systems, we find the above assumption fulfilled for all examples considered. Through an analysis based on Fermi's golden rule, we partially relate these findings to the applicability of the eigenstate thermalization ansatz to the respective driving operators.

Quantum-to-classical transition in the work distribution for chaotic systems

The work distribution is a fundamental quantity in nonequilibrium thermodynamics mainly due to its connection with fluctuation theorems. Here, we develop a semiclassical approximation to the work distribution for a quench process in chaotic systems that provides a link between the quantum and classical work distributions. The approach is based on the dephasing representation of the quantum Loschmidt echo and on the quantum ergodic conjecture, which states that the Wigner function of a typical eigenstate of a classically chaotic Hamiltonian is equidistributed on the energy shell. Using numerical simulations, we show that our semiclassical approximation accurately describes the quantum distribution as the temperature is increased.

Quantum Bochkov-Kuzovlev work fluctuation theorems

The quantum version of the Bochkov-Kuzovlev identity is derived on the basis of the appropriate definition of work as the difference of the measured internal energies of a quantum system at the beginning and the end of an external action on the system given by a prescribed protocol. According to the spirit of the original Bochkov-Kuzovlev approach, we adopt the 'exclusive' viewpoint, meaning that the coupling to the external work source is not counted as part of the internal energy. The corresponding canonical and microcanonical quantum fluctuation theorems are derived as well, and are compared with the respective theorems obtained within the 'inclusive' approach. The relations between the quantum inclusive work w, the exclusive work w(0) and the dissipated work w(dis), are discussed and clarified. We show by an explicit example that w(0) and w(dis) are distinct stochastic quantities obeying different statistics.

Finite bath fluctuation theorem

We demonstrate that a finite bath fluctuation theorem of the Crooks type holds for systems that have been thermalized via weakly coupling them to a bath with energy independent finite specific heat. We show that this theorem reduces to the known canonical and microcanonical fluctuation theorems in the two respective limiting cases of infinite and vanishing specific heat of the bath. The result is elucidated by applying it to a two-dimensional hard disk colliding elastically with few other hard disks in a rectangular box with perfectly reflecting walls.