Monte Carlo simulations of Ising models coupled to heat baths at two different temperatures are used to study a fluctuation relation for the heat exchanged between the two thermostats in a time $τ$. Different kinetics (single--spin--flip or spin--exchange Kawasaki dynamics), transition rates (Glauber or Metropolis), and couplings between the system and the thermostats have been considered. In every case the fluctuation relation is verified in the large $τ$ limit, both in the disordered and in the low temperature phase. Finite-$τ$ corrections are shown to obey a scaling behavior.
We derive a direct connection between Monte Carlo time and physical time in terms of physical parameters, using a quantum Hamiltonian with a $d$-dimensional phonon heat bath interacting with a square-lattice Ising ferromagnet. Based on the calculated transition rates, we perform dynamic Monte Carlo simulations using absorbing Markov chains to measure the lifetimes of the metastable state at low temperatures. We also calculate the lifetimes analytically using absorbing Markov chains. The phonon dynamic gives field-dependent prefactors in the lifetimes at low temperatures, that are different fro…
The Ising model doesn't have a strictly defined dynamics, only a spectrum. There are different ways to equip it with a time dependence e.g. the Glauber or the Kawasaki dynamics, which are both stochastic, but it means there is a master equation which can also describes their dynamics. We present a Gluber-type master equation derived from the Redfield equation, where the spin system is coupled to a bosonic bath. We derive a time dependent mean field equation which describes the relaxation of the spin system at finite temperature. Using the fully connected, uniform Ising model the relaxation tim…
Starting from the Liouville-von Neumann equation, under a weak coupling limit we derive the Lindblad master equation for the one-dimensional quantum Ising model in a Markov approximation and a rotating wave approximation. We also prove that the steady solution of the Lindblad equation is the canonical distribution independent of the dissipation rate.
Based on the Robertson theory the nonlinear dynamics of general Ising systems coupled microscopically to bath systems is investigated leading to two complimentary approaches. Within the master equation approach microscopically founded transition rates are presented which essentially differ from the usual phenological rates. The second approach leads to coupled equations of motion for the local magnetizations and the exchange energy. Simple examples are discussed and the general results are applied to the Sherrington-Kirkpatrick spin glass model.
We compute the critical exponents associated with a magnetic line defect in the critical 3D Ising model. From the result, we deduce the anomalous dimension of the fermion operator in the z = 1 scaling regime of a Fermi surface coupled to a O(1) Wilson-Fisher fixed point. We introduce a generalization of the Wolff cluster algorithm to O(N) spin systems with a background magnetic field.
When a small dynamical system that is initially in contact with a heat bath is detached from this heat bath and then caused to undergo a quasi-static adiabatic processes, the statistical distribution of the system's energy differs from that of an equilibrium ensemble. Subsequent contact of the system with another heat bath is inevitably irreversible, hence the entire process cannot be reversed without a net energy transfer to the heat baths.
Fluctuation relations for the entropy production in non equilibrium stationary states of Ising models are investigated by Monte Carlo simulations. Systems in contact with heat baths at two different temperatures or subject to external driving will be studied. In the first case, by considering different kinetic rules and couplings with the baths, the behavior of the probability distributions of the heat exchanged in a time $τ$ with the thermostats, both in the disordered and in the low temperature phase, are discussed. The fluctuation relation is always verified in the large $τ$ limit and devia…
We introduce new fast canonical local algorithms for discrete and continuous spin systems. We show that for a broad selection of spin systems they compare favorably to the known ones except for the Ising +/-1 spins. The new procedures use discretization scheme and the necessary information have to be stored in computer memory before the simulation. The models for testing discrete spins are the Ising +/-1, the general Ising S or Blume-Capel model, the Potts and the clock models. The continuous spins we examine are the O(N) models, including the continuous Ising model (N=1), the ϕ^4 Ising model …
We study the fate of the Ising model and its universal properties when driven by a rapid periodic drive and weakly coupled to a bath at equilibrium. The far from equilibrium steady-state regime of the system is accessed by means of a Floquet mean field approach. We show that, depending on the details of the bath, the drive can strongly renormalize the critical temperature to higher temperatures, modify the critical exponents, or even change the nature of the phase transition from second to first order after the emergence of a tricritical point. Moreover, by judiciously selecting the frequency …
The Carnot theorem, one expression of the second law of thermodynamics, places a fundamental upper bound on the efficiency of heat engines operating between two heat baths. The Carnot theorem can be stated in a more generalized form for heat engines operating with multiple heat baths, where the maximum efficiency is achieved for reversible heat engines operating quasistatically between two heat baths. In this study, we determine the irreversible efficiency of heat engines operating with multiple heat baths in a linear response regime, i.e., under small temperature differences and a slow variat…
Nonlinear refrigerator with a finite-sized cold heat bath
The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Division of Sciences, Krea University（克里亚大学理学院）；Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo（东京大学前沿科学研究生院复杂科学与工程系）
We analyze the stability of a gyroscopic oscillator interacting with a finite- and infinite-dimensional heat bath in both the classical and quantum cases. We consider a finite gyroscopic oscillator model of a particle in a magnetic field and examine the stability before and after coupling to a heat bath. It is shown that if the oscillator is gyroscopically stable, coupling to a sufficiently massive heat bath induces instability. The meaning of these ideas in the quantum context is discussed. The model extends the exact diagonalization analysis of an oscillator and field of Ford, Lewis, and O'C…
In [Phys. Rev. E 88, 042126 (2013)] it is stated that Tsallis distributions do not emerge from thermalization with a "bath" of finite, energy-independent, heat capacity. We report evidence for the contrary.
We examine a square-lattice nearest-neighbor Ising quantum ferromagnet coupled to $d$-dimensional phonon baths. Using the density-matrix equation, we calculate the transition rates between configurations, which determines the specific dynamic. Applying the calculated stochastic dynamic in Monte Carlo simulations, we measure the lifetimes of the metastable state. As the magnetic field approaches $|H|/J=2$ at low temperatures, the lifetime prefactor diverges because the transition rates between certain configurations approaches zero under these conditions. Near $|H|/J=2$ and zero temperature, th…
We measure the energy fluctuations of a Brownian particle confined by an optical trap in an aging gelatin after a very fast quench (less than 1 ms). The strong nonequilibrium fluctuations due to the assemblage of the gel, are interpreted, within the framework of fluctuation theorem, as a heat flux from the particle towards the bath. We derive, from a simple model, an analytical expression of the heat probability distribution, which fits the experimental data and satisfies a fluctuation relation similar to that of a system in contact with two baths at different temperatures.
We investigate three kinds of heat produced in a system and a bath strongly coupled via an interaction Hamiltonian. By studying the energy flows between the system, the bath, and their interaction, we provide rigorous definitions of two types of heat, $Q_{\rm S}$ and $Q_{\rm B}$ from the energy loss of the system and the energy gain of the bath, respectively. This is in contrast to the equivalence of $Q_{\rm S}$ and $Q_{\rm B}$, which is commonly assumed to hold in the weak coupling regime. The bath we consider is equipped with a thermostat which enables it to reach an equilibrium. We identify…
We consider a system in a non-equilibrium steady state by joining two semi-infinite Ising chains coupled to thermal reservoirs with {\em different} temperatures, $T$ and $T^{\prime}$. To compute the energy flux from the hot bath through our system into the cold bath, we exploit Glauber heat-bath dynamics to derive an exact equation for the two-spin correlations, which we solve for $T^{\prime}=\infty$ and arbitrary $T$. We find that, in the $T'=\infty$ sector, the in-flux occurs only at the first spin. In the $T<\infty$ sector (sites $x=1,2,...$), the out-flux shows a non-trivial profile: $F(x)…
We study relaxational behavior from a highly excited state for a composite system in partial contact with a heat bath, motivated by an experimental report of long-term energy storage in protein molecules. The system consists of two coupled elements: The first element is in direct contact with a heat bath, while the second element interacts only with the first element. Due to this indirect contact with the heat bath, energy injected into the second element dissipates very slowly, according to a power law, whereas that injected into the first one exhibits exponential dissipation. The relaxation …
We obtain an exact formula for the equilibrium free energy of a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a heat bath of independent quantum harmonic oscillators through the momentum variables. We show that the free energy has a different expression than that for the coordinate-coordinate coupling between the particle and the heat-bath oscillators. For an illustrative heat-bath spectrum, we evaluate the free energy in the low-temperature limit, thereby showing that the entropy of the charged particle van…
We study the dissipative dynamics of a harmonic oscillator which couples linearly through its position and its momentum to two independent heat baths at the same temperature. We argue that this model describes a large spin in a ferromagnet. We find that some effects of the two heat baths partially cancel each other. This leads to unexpected features such as underdamped oscillations and long relaxation times in the strong coupling regime. Such a partial frustration of dissipation can be ascribed to the canonically conjugate character of position and momentum. We compare this model to the scenar…
We performed numerical simulations on a two-dimensional inelastic hard disk system under gravity with a heat bath to study the dynamics of granular fluidization. Upon increasing the temperature of the heat bath, we found that three phases, namely, the condensed phase, locally fluidized phase, and granular turbulent phase, can be distinguished using the maximum packing fraction and the excitation ratio, or the ratio of the kinetic energy to the potential energy.It is shown that the system behavior in each phase is very different from that of an ordinary vibrating bed.
Important models of nonequilibrium statistical physics (NESP) are limited by a commonly used, but often unrecognized, near-equilibrium approximation. Fokker-Planck and Langevin equations, the Einstein and random-flight diffusion models, and the Schnakenberg model of biochemical networks suppose that fluctuations are due to an ideal equilibrium bath. But far from equilibrium, this perfect bath concept does not hold. A more principled approach should derive the rate fluctuations from an underlying dynamical model, rather than assuming a particular form. Here, using Maximum Caliber as the underly…
On the basis of analytical results and molecular dynamics simulations we clarify the nonequilibrium dynamics of a long-range interacting system in contact with a heat bath. For small couplings with the bath, we show that the system can first be trapped in a Vlasov quasi-stationary state, then a microcanonical one follows, and finally canonical equilibrium is reached at the bath temperature. We compare a Langevin mesoscopic thermostat with Hamiltonian reservoirs microscopically coupled with the system and demonstrate the equivalence of the two descriptions. Our identification of the parameters …
We derive the thermal conductivities of one-dimensional harmonic and anharmonic lattices with self-consistent heat baths (BRV lattice) from the Single-Mode Relaxation Time (SMRT) approximation. For harmonic lattice, we obtain the same result as previous works. However, our approach is heuristic and reveals phonon picture explicitly within the heat transport process. The results for harmonic and anharmonic lattices are compared with numerical calculations from Green-Kubo formula. The consistency between derivation and simulation strongly supports that effective (renormalized) phonons are energy…
Using a time-averaging technique we obtain exactly the probability distribution for position and velocity of a Brownian particle under the influence of two heat baths at different temperatures. These baths are expressed by a white noise term, representing the fast dynamics, and a colored noise term, representing the slow dynamics. Our exact solution scheme accounts for inertial effects, that are not present in approaches that assume the Brownian particle in the over-damped limit. We are also able to obtain the contribution associated with the fast noise that are usually neglected by other appr…
It is often stated that heat baths with finite degrees of freedom i.e., finite baths, are sources of Tsallis distributions for the classical Hamiltonian systems. By using well-known fundamental statistical mechanical expressions, we rigorously show that Tsallis distributions with fat tails are possible \textit{only} for finite baths with constant negative heat capacity while constant positive heat capacity finite baths yield decays with sharp cut-off with no fat tails. However, the correspondence between Tsallis distributions and finite baths holds at the expense of violating equipartition the…
Thermodynamics of nanoscale devices is an active area of research. Despite their noisy surround- ing they often produce mechanical work (e.g. micro-heat engines) or display rectified Brownian motion (e.g. molecular motors). This invokes the research in terms of experimentally quantifiable thermodynamic efficiencies. To enhance the efficiency of such devices, close-loop control is an useful technique. Here a single Brownian particle is driven by a harmonic confinement with time-periodic contraction and expansion, together with a velocity feedback that acts on the particle only when the trap con…
Work can be extracted from a single heat bath if additional information is available. For the paradigmatic case of a Brownian particle in a harmonic potential, whose position has been measured with finite precision, we determine the optimal protocol for manipulating the center and stiffness of the potential in order to maximize this work in a finite-time process. The bound on this work imposed by a generalized second law inequality involving information can be reached only if both position and stiffness of the potential are controlled and the process is quasistatic. Estimates on the power deli…
A unified $χ$-criterion for heat devices (including heat engines and refrigerators) which is defined as the product of the energy conversion efficiency and the heat absorbed per unit time by the working substance [de Tomás \emph{et al} 2012 \textit{Phys. Rev. E} \textbf{85} 010104(R)] is optimized for tight-coupling heat engines and refrigerators operating between two heat baths at temperatures $T_c$ and $T_h(>T_c)$. By taking a new convention on the thermodynamic flux related to the heat transfer between two baths, we find that for a refrigerator tightly and symmetrically coupled with two hea…
We study the Quantum Brownian motion of a charged particle moving in a harmonic potential in the presence of an uniform external magnetic field and linearly coupled to an Ohmic bath through momentum variables. We analyse the growth of the mean square displacement of the particle in the classical high temperature domain and in the quantum low temperature domain dominated by zero point fluctuations. We also analyse the Position Response Function and the long time tails of various correlation functions. We notice some distinctive features, different from the usual case of a charged quantum Browni…
We study a Langevin-like model which describes an inertial particle in a one-dimensional harmonic potential and subjected to two heat baths and one athermal environment. The thermal noises are white and Gaussian, and the temperatures of heat reservoirs are different. The athermal medium act through an external non-Gaussian noise of Poisson type. We calculate exactly the time-dependent cumulant-generating function of position and velocity of the particle, as well as an expression of this generating function for stationary states. We discuss the long-time behavior of first cumulants of the energ…
We apply the concept of a frequency-dependent effective temperature based on the fluctuation-dissipation ratio to a driven Brownian particle in a nonequilibrium steady state. Using this system as a thermostat for a weakly coupled harmonic oscillator, the oscillator thermalizes according to a canonical distribution at the respective effective temperature across the entire frequency spectrum. By turning the oscillator from a passive "thermometer" into a heat engine, we realize the cyclic extraction of work from a single thermal reservoir, which is feasible only due to its nonequilibrium nature.
We study quantum annealing of pure and random transverse Ising chains that are coupled to boson baths, using a novel numerical method based on the combination of the quasi-adiabatic propagation path integral (QUAPI) and the matrix product state (MPS) formalisms. We present numerical results on systems up to 64 spins and conclude that the baths disturb quantum annealing of pure and random Ising chains. The numerical method to compute the reduced density matrix is described in detail.
We consider a situation where an $N$-level system (NLS) is coupled successively to two heat baths with different temperatures without being necessarily thermalized and approaches a steady state. For this situation we apply a general Jarzinski-type equation and conclude that heat and entropy is flowing from the hot bath to the cold one. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities. Our approach is illustrated by an analytical example. For the linear regime, i.~e., for small temperature differences b…
We consider a situation where an $N$-level system (NLS) is coupled to a heat bath without being necessarily thermalized. For this situation we derive general Jarzinski-type equations and conclude that heat and entropy is flowing from the hot bath to the cold NLS and, vice versa, from the hot NLS to the cold bath. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities which have already been considered in the literature. Our approach is illustrated by an analytical example.
This paper is about quantum heat defined as the change in energy of a bath during a process. The presentation takes into account recent developments in classical strong-coupling thermodynamics, and addresses a version of quantum heat which satisfies quantum-classical correspondence. The characteristic function and the full counting statistics of quantum heat are shown to be formally similar. The paper further shows that the method can be extended to more than one bath, e.g. two baths at different temperatures, which opens up the prospect of studying correlations and heat flow. The paper extend…
A class of autonomous quantum heat baths satisfying the eigenstate thermalization hypothesis (ETH) criteria is proposed. We show that such systems are expected to cause thermal relaxation of much smaller quantum systems coupled to one of the baths local observables. The process of thermalization is examined through residual fluctuations of local observables of the bath around their thermal values predicted by ETH. It is shown that such fluctuations perturb the small quantum system causing its decoherence to the thermal state. As an example, we investigate theoretically and numerically thermali…
We show that the case of two oscillators in a common heat bath cannot be reduced to an effective one body problem. In addition, there is an interaction between the oscillators, even at zero temperature, due to the fluctuations caused in both oscillators by the zero-point oscillations of the electromagnetic field.
We consider a quantum system strongly coupled to multiple heat baths at different temperatures. Quantum heat transport phenomena in this system are investigated using two definitions of the heat current, one in terms of the system energy, and the other in terms of the bath energy. When we consider correlations among system-bath interactions (CASBI) -- which have a purely quantum mechanical origin -- the definition in terms of the bath energy becomes different. We found that CASBI are necessary to maintain the consistency of the heat current with thermodynamic laws in the case of strong system-…
In this paper we investigate the relationship between the efficiency of a cyclic quantum heat engine with the Hilbert space dimension of the thermal baths. By means of a general inequality, we show that the Carnot efficiency can be obtained only when both the hot and cold baths are infinitely large. By further introducing a specific model where the baths are constituted of ensembles of finite-dimensional particles, we further demonstrate the relationship between the engine's power and efficiency, with the dimension of the working substance and the bath particles.
We provide an explicit method for solving general markovian master equations for quadratic bosonic Hamiltonians with linear bath operators. As an example we consider a one-dimensional quantum harmonic oscillator chain coupled to thermal reservoirs at both ends of the chain. We derive an analytic solution of the Redfield master equation for homogeneous harmonic chain and recover classical results, namely, vanishing temperature gradient and constant heat current in the thermodynamic limit. In the case of the disordered gapped chains we observe universal heat current scaling independent of the ba…
Based on the system heat bath approach where the bath is nonlinearly modulated by an external Gaussian random force, we propose a new microscopic model to study directed motion in the overdamped limit for a nonequilibrium open system. Making use of the coupling between the heat bath and the external modulation as a small perturbation we construct a Langevin equation with multiplicative noise and space dependent dissipation and the corresponding Fokker-Planck-Smoluchowski equation in the overdamped limit. We examine the thermodynamic consistency condition and explore the possibility of observin…
The optimal efficiency of quantum (or classical) heat engines whose heat baths are $n$-particle systems is given by the information geometry and the strong large deviation. We give the optimal work extraction process as a concrete energy-preserving unitary time evolution among the heat baths and the work storage. We show that our optimal work extraction turns the disordered energy of the heat baths to the ordered energy of the work storage, by evaluating the ratio of the entropy difference to the energy difference in the heat baths and the work storage, respectively. By comparing the statistic…
We determine the zero and finite temperature phase diagram of the fully frustrated quantum Ising model on the bathroom tile (4-8) lattice. The phase diagram exhibits a wealth of 2+1d physics, including 1. classical Coulomb dimer liquids of both square and triangular lattice types; 2. quantum order-by-disorder induced phases breaking $\mathbb{Z}_4$, $\mathbb{Z}_6$, and $\mathbb{Z}_8$ symmetries; 3. finite temperature Kosterlitz-Thouless (KT) phases floating over the $\mathbb{Z}_6$ and $\mathbb{Z}_8$ orders; and, 4. staircases of (in)-commensurate symmetry breaking phases at intermediate couplin…
It has recently been shown that in one-dimensional hard-point gases, there is a mechanism that induces negative differential thermal resistance (NDTR) between heat baths. We examine this mechanism in more general higher dimensional fluids described by multiparticle collision dynamics. We consider fluids in a finite cuboid region of three-dimensional space with each end in contact with a heat bath. Based on analytical results and numerical models, we find that the mechanism underlying NDTR also works for high-dimensional fluidic systems with weak interactions and is very robust to mixed fluids.…
We illustrate for 4D SU(2) and U(1) lattice gauge theory that sampling with a biased Metropolis scheme is essentially equivalent to using the heat bath algorithm. Only, the biased Metropolis method can also be applied when an efficient heat bath algorithm does not exist.
We discuss thermalization of a test particle schematized as a harmonic oscillator and coupled to a Boltzmann heat bath of finite size and with a finite bandwidth for the frequencies of its particles. We find that complete thermalization only occurs when the test particle frequency is within a certain range of the bath particle frequencies, and for a certain range of mass ratios between the test particle and the bath particles. These results have implications for the study of classical and quantum behaviour of high-frequency nanomechanical resonators.
We discuss the problem of heat conduction in quantum spin chain models. To investigate this problem it is necessary to consider the finite open system connected to heat baths. We describe two different procedures to couple the system with the reservoirs: a model of stochastic heat baths and the quantum trajectories solution of the quantum master equation. The stochastic heat bath procedure operates on the pure wave function of the isolated system, so that it is locally and periodically collapsed to a quantum state consistent with a boundary nonequilibrium state. In contrast, the quantum trajec…
We investigate the transport of phonons between $N$ harmonic oscillators in contact with independent thermal baths and coupled to a common oscillator, and derive an expression for the steady state heat flow between the oscillators in the weak coupling limit. We apply these results to an optomechanical array consisting of a pair of mechanical resonators coupled to a single quantised electromagnetic field mode by radiation pressure as well as to thermal baths with different temperatures. In the weak coupling limit this system is shown to be equivalent to two mutually-coupled harmonic oscillators…
Small systems (of interest in the areas of nanophysics, quantum information, etc.) are particularly vulnerable to environmental effects. Thus, we determine various thermodynamic functions for an oscillator in an arbitrary heat bath at arbitrary temperatures. Explicit results are presented for the most commonly discussed heat bath models: Ohmic, single relaxation time and blackbody radiation.
In this paper we consider thermal power of a heat flow through a qubit between two baths. The baths are modeled as set of harmonic oscillators initially at equilibrium, at two temperatures. Heat is defined as the change of energy of the cold bath, and thermal power is defined as expected heat per unit time, in the long-time limit. The qubit and the baths interact as in the spin-boson model, i.e. through qubit operator $σ_z$. We compute thermal power in an approximation analogous to `non-interacting blip' (NIBA) and express it in the polaron picture as products of correlation functions of the t…
We study thermodynamic processes in contact with a heat bath that may have an arbitrary time-varying periodic temperature profile. Within the framework of stochastic thermodynamics, and for models of thermo-dynamic engines in the idealized case of underdamped particles in the low-friction regime, we derive explicit bounds as well as optimal control protocols that draw maximum power and achieve maximum efficiency at any specified level of power.
Thermodynamics has recently been extended to small scales with resource theories that model heat exchanges. Real physical systems exchange diverse quantities: heat, particles, angular momentum, etc. We generalize thermodynamic resource theories to exchanges of observables other than heat, to baths other than heat baths, and to free energies other than the Helmholtz free energy. These generalizations are illustrated with "grand-potential" theories that model movements of heat and particles. Free operations include unitaries that conserve energy and particle number. From this conservation law an…
To understand the mechanism allowing for the long-term storage of excess energy in proteins, we study a Hamiltonian system consisting of several coupled pendula in partial contact with a heat bath. It is found that energy storage is possible when the motion of each pendulum switches between oscillatory (vibrational) and rotational (phase-slip) modes. The storage time increases almost exponentially to the square root of the injected energy. The relevance of our mechanism to protein motors is discussed.
We study the heat and spin transport properties in a ring of interacting spins coupled to heat baths at different temperatures. We show that interactions, by inducing avoided crossings, can be a mean to tune both the total heat current flowing between the ring and the baths, and the way it flows through the system. In particular, we recognize three regimes in which the heat current flows clockwise, counter-clockwise, and in parallel. The temperature bias between the baths also induces a spin current within the ring, whose direction and magnitude can be tuned by the interaction. Lastly, we show…
We consider a one-dimensional Ising model in a transverse magnetic field coupled to a dissipative heat bath. The phase diagram and the critical exponents are determined from extensive Monte Carlo simulations. It is shown that the character of the quantum phase transition is radically altered from the corresponding non-dissipative model and the double-well coupled to a dissipative heat bath with linear friction. Spatial couplings and the dissipative dynamics combine to form a new quantum criticality.
