We have studied the specific heat of the (NS + NB) model for an NS-body harmonic oscillator (HO) system which is strongly coupled to an NB-body HO bath without dissipation. The system specific heat of CS(T) becomes NSkB at T → ∞ and vanishes at T = 0 in accordance with the third law of thermodynamics. The calculated CS(T) at low temperatures is not proportional to NS and shows an anomalous temperature dependence, strongly depending on NS, NB, and the system-bath coupling. In particular at very low (but finite) temperatures, it may become negative for a strong system-bath coupling, which is in contrast with non-negative specific heat of a HO system with NS = 1 reported by Ingold, Hänggi, and Talkner [Phys. Rev. E 79, 061105 (2005)]. Our calculation indicates an importance of taking account of finite NS in studying open quantum systems which may include an arbitrary number of particles in general.
We have studied responses to applied external forces of the quantum
(NS+NB) model for
NS-body interacting harmonic oscillator (HO) system subjected to
NB-body HO bath, by using canonical transformations combined with Husimi's method for a driven quantum HO [K. Husimi, Prog. Theor. Phys. {\bf 9}, 381 (1953)]. It has been shown that the response to a uniform force expressed by the Hamiltonian:
Hf=−f(t)∑k=1NSQk is generally not proportional to
NS except for no system-bath couplings, where
f(t) expresses its time dependence and
Qk denotes a position operator of
kth particle of the system. We have calculated also the response to a space- and time-dependent force expressed by
Hf=−f(t)∑k=1NSQkei2πku/NS, where the wavevector
u is
u=0 and
u=−NS/2 for uniform and staggered forces, respectively. The spatial correlation
Γm for a pair of positions of
Qk and
Qk+m has been studied as functions of
NS and the temperature. Our calculations have indicated an importance of taking account of finite
NS in studying quantum open systems which generally include arbitrary numbers of particles.
The evaluation of the specific heat of an open damped quantum system is a subtle issue. One possible route is based on the thermodynamic partition function which is the ratio of the partition functions of system plus bath and of the bath alone. For the free damped particle it has been shown, however, that the ensuing specific heat may become negative for appropriately chosen environments. Being an open system this quantity then naturally must be interpreted as the change in the specific heat obtained as the difference between the specific heat of the heat bath coupled to the system degrees of freedom and the specific heat of the bath alone. While this difference may become negative, the involved specific heats themselves are always positive; thus, the known thermodynamic stability criteria are perfectly guaranteed. For a damped quantum harmonic oscillator, instead of negative values, under appropriate conditions one can observe a dip in the difference of specific heats as a function of temperature. Stylized minimal models containing a single oscillator heat bath are employed to elucidate the occurrence of the anomalous temperature dependence of the corresponding specific heat values. Moreover, we comment on the consequences for the interpretation of the density of states based on the thermal partition function.
We consider a free particle coupled with finite strength to a bath and investigate the evaluation of its specific heat. A harmonic oscillator bath of Drude type with cutoff frequency ω D is employed to model an ohmic friction force with dissipation strength γ. Two scenarios for obtaining specific heat are presented. The first one uses the measurement of the kinetic energy of the free particle, while the second one is based on the reduced partition function. Both descriptions yield results which are consistent with the Third Law of thermodynamics. Nevertheless, the two methods produce different results that disagree even in their leading quantum corrections at high temperatures. We also consider the regime where the cutoff frequency is smaller than the friction strength, i.e. ω D < γ. There, we encounter puzzling results at low temperatures where the specific heat based on the thermodynamic prescription becomes negative. This anomaly is rooted in an ill-defined density of states of the damped free particle, which assumes unphysical negative values when γ/ω D > 1.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
The prevailing method for addressing strong-coupling thermodynamics typically involves open quantum systems coupled to infinite baths in equilibrium through the Hamiltonian of mean force (HMF). However, its applicability to finite baths remains limited. In this work, we transcend this limit by considering the impacts of system-bath coupling which is not only on the system but also on the finite bath feedback. When the bath and system sizes are comparable, they act as each other's effective 'bath'. We introduce a parameter α within [0,1], where α and 1-α act as weighting factors to distribute the system-bath coupling between the system and the bath. Our approach innovatively incorporates the effect of coupling on the respective effective 'bath' into both the system and bath in the statistical factor form. We generalize the HMF and propose quantum Hamiltonians of mean forces to handle the distributed coupling of the system to finite baths. The additional parameter α is determined by minimizing the joint free energy density and is a new thermodynamic quantity representing the finite bath's influence. Examining the damped quantum harmonic oscillator with a finite bath, we find that α affects both states, leading to a complex phase diagram with unique phenomena at critical temperatures T_{L} and T_{R}, including valleys, peaks, negative values, and discontinuities in entropy and specific heat. Notably, these anomalies disappear when α→1, both at high temperatures and in the thermodynamic limit, indicating the negligible influence of coupling on the bath states and reverting to the HMF. Thus, studying finite-sized baths holds substantial significance in small quantum systems.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.