The positivity of the heat capacity is the hallmark of thermal stability of systems in thermodynamic equilibrium. We show that this property remains valid for systems with negative derivative of energy with respect to temperature, as happens to some system described by the microcanonical ensemble. The demonstration rests on considering a trajectory on the Gibbs equilibrium surface, and its projection on the entropy-energy plane. The Gibbs equilibrium surface has the convexity property, but the projection might lack this property, leading to a negative derivative of energy with respect to temperature.
We develop a scaling theory for the finite-size critical behavior of the microcanonical entropy (density of states) of a system with a critically-divergent heat capacity. The link between the microcanonical entropy and the canonical energy distribution is exploited to establish the former, and corroborate its predicted scaling form, in the case of the 3d Ising universality class. We show that the scaling behavior emerges clearly when one accounts for the effects of the negative background constant contribution to the canonical critical specific heat. We show that this same constant plays a sig…
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 …
For models which exhibit a continuous phase transition in the thermodynamic limit a numerical study of small systems reveals a non-monotonic behaviour of the microcanonical specific heat as a function of the system size. This is in contrast to a treatment in the canonical ensemble where the maximum of the specific heat increases monotonically with the size of the system. A phenomenological theory is developed which permits to describe this peculiar behaviour of the microcanonical specific heat and allows in principle the determination of microcanonical critical exponents.
A microcanonical first order transition, connecting a clustered to a homogeneous phase, is studied from both the thermodynamic and dynamical point of view for a N-body Hamiltonian system with infinite-range couplings. In the microcanonical ensemble specific heat can be negative, but besides that, a microcanonical first order transition displays a temperature discontinuity as the energy is varied continuously (a dual phenomenon to the latent heat in the canonical ensemble). In the transition region, the entropy per particle exhibits, as a function of the order parameter, two relative maxima sep…
A Microcanonical Finite Site Ansatz in terms of quantities measurable in a Finite Lattice allows to extend phenomenological renormalization (the so called quotients method) to the microcanonical ensemble. The Ansatz is tested numerically in two models where the canonical specific-heat diverges at criticality, thus implying Fisher-renormalization of the critical exponents: the 3D ferromagnetic Ising model and the 2D four-states Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows to simulate syst…
The canonical ensemble describes an open system in equilibrium with a heat bath of fixed temperature. The probability distribution of such a system, the Boltzmann distribution, is derived from the uniform probability distribution of the closed universe consisting of the open system and the heat bath, by taking the limit where the heat bath is much larger than the system of interest. Alternatively, the Boltzmann distribution can be derived from the Maximum Entropy Principle, where the Gibbs-Shannon entropy is maximized under the constraint that the mean energy of the open system is fixed. To ma…
By using projection superoperators, we present a new derivation of the quantum master equation first obtained by the Authors in Phys. Rev. E {\bf 68}, 066112 (2003). We show that this equation describes the dynamics of a subsystem weakly interacting with an environment of finite heat capacity and initially described by a microcanonical distribution. After applying the rotating wave approximation to the equation, we show that the subsystem dynamics preserves the energy of the total system (subsystem plus environment) and tends towards an equilibrium state which corresponds to equipartition insi…
We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different mi…
We calculate exactly both the microcanonical and canonical thermodynamic functions (TDFs) for a one-dimensional model system with piecewise constant Lennard-Jones type pair interactions. In the case of an isolated $N$-particle system, the microcanonical TDFs exhibit (N-1) singular (non-analytic) microscopic phase transitions of the formal order N/2, separating N energetically different evaporation (dissociation) states. In a suitably designed evaporation experiment, these types of phase transitions should manifest themselves in the form of pressure and temperature oscillations, indicating cool…
The main interest of the present work is the generalization of the Boltzmann-Gibbs distributions and the fluctuation theory based on the consideration of the reparametrization invariance of the microcanonical ensemble. This approach allows a novel interpretation of some anomalous phenomena observed in the non extensive systems like the existence of the negative specific heats as well as possibilities the enhancing of some Monte Carlo methods based on the Statistical Mechanics.
By using freeze-out properties of multifragmenting hot nuclei produced in quasifusion central $^{129}$Xe+$^{nat}$Sn collisions at different beam energies (32, 39, 45 and 50 AMeV) which were estimated by means of a simulation based on experimental data collected by the $4π$ INDRA multidetector, heat capacity in the thermal excitation energy range 4 - 12.5 AMeV was calculated from total kinetic energies and multiplicities at freeze-out. The microcanonical formulation was employed. Negative heat capacity which signs a first order phase transition for finite systems is observed and confirms previo…
We study the warming process of a semi-infinite cylindrical Ising lattice initially ordered and coupled at the boundary to a heat reservoir. The adoption of a proper microcanonical dynamics allows a detailed study of the time evolution of the system. As expected, thermal propagation displays a diffusive character and the spatial correlations decay exponentially in the direction orthogonal to the heat flow. However, we show that the approach to equilibrium presents an unexpected slow behavior. In particular, when the thermostat is at infinite temperature, correlations decay to their asymptotic …
For a macroscopic, isolated quantum system in an unknown pure state, the expectation value of any given observable is shown to hardly deviate from the ensemble average with extremely high probability under generic equilibrium and nonequilibrium conditions. Special care is devoted to the uncontrollable microscopic details of the system state. For a subsystem weakly coupled to a large heat bath, the canonical ensemble is recovered under much more general and realistic assumptions than those implicit in the usual microcanonical description of the composite system at equilibrium.
When studying the thermodynamic properties of mesoscopic systems the most appropriate microcanonical entropy is the volume entropy, i.e. the logarithm of the volume of phase space enclosed by the hypersurface of constant energy. For systems with broken ergodicity, the volume entropy has discontinuous jumps at values of energy that correspond to separatrix trajectories. Simultaneously there is a convex intruder in the entropy function and a region of negative specific heat below such critical energies. We illustrate this with a simple model composed of a chain of 3 particles which interact via …
We investigate the limiting cases of Tsallis statistics. The viewpoint adopted is not the standard information-theoretic one, where one derives the distribution from a given measure of information. Instead the mechanical approach recently proposed in [M. Campisi, G.B. Bagci, Phys. Lett. A (2006), doi:10.1016/j.physleta.2006.09.081], is adopted, where the distribution is given and one looks for the associated physical entropy. We show that, not only the canonical ensemble is recovered in the limit of $q$ tending to one, as one expects, but also the microcanonical ensemble is recovered in the li…
Heat capacities of model systems with finite numbers of effective degrees of freedom are evaluated using canonical and microcanonical thermodynamics. Discrepancies between both approaches, which are observed even in the infinite-size limit, are particularly large in systems that exhibit an excited-state quantum phase transition. The corresponding irregularity of the spectrum generates a singularity in the microcanonical heat capacity and affects smoothly the canonical heat capacity.
Following recent work by Marathe and Parrondo [PRL, 104, 245704 (2010)], we construct a classical Hamiltonian system whose energy is reduced during the adiabatic cycling of external parameters, when initial conditions are sampled microcanonically. Combining our system with a device that measures its energy, we propose a cyclic procedure during which energy is extracted from a heat bath and converted to work, in apparent violation of the second law of thermodynamics. This paradox is resolved by deriving an explicit relationship between the average work delivered during one cycle of operation, a…
Thermodynamics allows the application of Statistical Mechanics to finite and even small systems. As surface effects cannot be scaled away, one has to be careful with the standard arguments of splitting a system into two or bringing two systems into thermal contact with energy or particle exchange: Not only the volume part of the entropy must be considered. The addition of any other macroscopic constraint like a dividing surface, or the enforcement of gradients of the energy/particle reduce the entropy. As will be shown here, when removing such constraint in regions of a negative heat capacity,…
It is shown that negative heat capacity in nanoclusters is an artifact of applying equilibrium thermodynamic formalism on a small system trapped out of equilibrium in a particular structural motif representing only part of the energetically available phase space volume. Trapping may occur in either the canonical or microcanonical ensemble, but it is unavoidable in the microcanonical. A more general consequence of trapping is that all macroscopic quantities determined for nanoclusters will depend on the initial conditions.
Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 170 years ago the original motivation of thermodynamics was the description of steam engines, i.e. boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by conventional thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang-Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann-Planck stati…
The site-diluted Ising model has been investigated using an improved microcanonical algorithm from Creutz Cellular Automaton. For a microcanonical algorithm, the basic problem is to estimate the correct temperatures using average values of the kinetic energy in the simulations of site-diluted Ising model. In this study, the average kinetic energy has been re-described with an expression dependent on dilution x=1-p. The values of the temperature have been calculated using the new expression and the critical temperatures have been estimated from the peaks of specific heat for each value of dilut…
In this work we extend the applicability of the microcanonical ensemble simulation method, originally proposed to study the Ising model (A. Hüller and M. Pleimling, Int. Journal of Modern Physics C, 13, 947 (2002), arxiv:cond-mat/0110090), to the case of simple fluids. An algorithm is developed by measuring the transition rates probabilities between macroscopic states, that has as advantage with respect to conventional Monte Carlo NVT (MC-NVT) simulations that a continuous range of temperatures are covered in a single run. For a given density, this new algorithm provides the inverse temperatur…
We study dipolarly coupled three dimensional spin systems in both the microcanonical and the canonical ensembles by introducing appropriate numerical methods to determine the microcanonical temperature and by realizing a canonical model of heat bath. In the microcanonical ensemble, we show the existence of a branch of stable antiferromagnetic states in the low energy region. Other metastable ferromagnetic states exist in this region: by externally perturbing them, an effective negative specific heat is obtained. In the canonical ensemble, for low temperatures, the same metastable states are un…
Microcanonical Thermodynamics allows the application of Statistical Mechanics on one hand to closed finite and even small systems and on the other to the largest,self-gravitating ones. However, one has to reconsider the fundamental principles of Statistical Mechanics especially its key quantity, entropy. Whereas in conventional Thermostatistics the homogeneity and extensivity of the system and the concavity of its entropy S(E) are central conditions, these fail for the systems considered here. E.g. at phase separation the entropy S(E) is necessarily convex to make e^{S(E)-E/T} bimodal in E (th…
We study the problem of ensemble equivalence in spin systems with short-range interactions under the existence of a first-order phase transition. The spherical model with nonlinear nearest-neighbour interactions is solved exactly both for canonical and microcanonical ensembles. The result reveals apparent ensemble inequivalence at the first-order transition point in the sense that the microcanonical entropy is non-concave as a function of the energy and consequently the specific heat is negative. In order to resolve the paradox, we show that an unconventional saddle point should be chosen in t…
We calculate the thermodynamic entropy of the mean-field $ϕ^4$ spin model in the microcanonical ensemble as a function of the energy and magnetization of the model. The entropy and its derivative are obtained from the theory of large deviations, as well as from Rugh's microcanonical formalism, which is implemented by computing averages of suitable observables in microcanonical molecular dynamics simulations. Our main finding is that the entropy is a concave function of the energy for all values of the magnetization, but is nonconcave as a function of the magnetization for some values of the en…
The anisotropic quantum Heisenberg model with Curie-Weiss-type interactions is studied analytically in several variants of the microcanonical ensemble. (Non)equivalence of microcanonical and canonical ensembles is investigated by studying the concavity properties of entropies. The microcanonical entropy s(e,m) is obtained as a function of the energy e and the magnetization vector m in the thermodynamic limit. Since, for this model, e is uniquely determined by m, the same information can be encoded either in s(m) or s(e,m1,m2). Although these two entropies correspond to the same physical settin…
Microcanonical thermodynamics allows the application of statistical mechanics both to finite and even small systems and also to the largest, self-gravitating ones. However, one must reconsider the fundamental principles of statistical mechanics especially its key quantity, entropy. Whereas in conventional thermostatistics, the homogeneity and extensivity of the system and the concavity of its entropy are central conditions, these fail for the systems considered here. For example, at phase separation, the entropy, S(E), is necessarily convex to make exp[S(E)-E/T] bimodal in E. Particularly, as …
Systems with long range interactions in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors. We propose a classification of microcanonical phase transitions, of their link to canonical ones, and of the possible situations of ensemble inequivalence. We discuss previously observed phase transitions and inequivalence in self-gravitating, two-dimensional fluid dynamics and non-neutral plasmas. We note a…
Velazquez and Curilef have proposed a methodology to extend Monte Carlo algorithms that are based on canonical ensemble. According to our previous study, their proposal allows us to overcome slow sampling problems in systems that undergo any type of temperature-driven phase transition. After a comprehensive review about ideas and connections of this framework, we discuss the application a re-weighting technique to improve the accuracy of microcanonical calculations, specifically, the well-known multi-histograms method of Ferrenberg and Swendsen. As example of application, we reconsider the stu…
The Blume-Capel model with infinite-range interactions presents analytical solutions in both canonical and microcanonical ensembles and therefore, its phase diagram is known in both ensembles. This model exhibits nonequivalent solutions and the microcanonical thermodynamical features present peculiar behaviors like nonconcave entropy, negative specific heat, and a jump in the thermodynamical temperature. Examples of nonequivalent ensembles are in general related to systems with long-range interactions that undergo canonical first-order phase transitions. Recently, the extended gaussian ensembl…
Kinetic energy fluctuations and specific heat in generalized ensembles
Research Center in the intersection of Plasma Physics, Matter and Complexity (P 2 mc), Comisión Chilena de Energía Nuclear（等离子体物理、物质与复杂性交叉研究中心，智利核能委员会）；Departamento de Física y Astronomía, Facultad de Ciencias Exactas, Universidad Andres Bello（安第斯贝尔纳多·乌尼韦斯德洛大学精确科学学院物理与天文学系）
We show that the projected Gross-Pitaevskii equation (PGPE) can be mapped exactly onto Hamilton's equations of motion for classical position and momentum variables. Making use of this mapping, we adapt techniques developed in statistical mechanics to calculate the temperature and chemical potential of a classical Bose field in the microcanonical ensemble. We apply the method to simulations of the PGPE, which can be used to represent the highly occupied modes of Bose condensed gases at finite temperature. The method is rigorous, valid beyond the realms of perturbation theory, and agrees with an…
We present a complete analytical solution of a system of Potts spins on a random k-regular graph in both the canonical and microcanonical ensembles, using the Large Deviation Cavity Method (LDCM). The solution is shown to be composed of three different branches, resulting in an non-concave entropy function.The analytical solution is confirmed with numerical Metropolis and Creutz simulations and our results clearly demonstrate the presence of a region with negative specific heat and, consequently, ensemble inequivalence between the canonical and microcanonical ensembles.
Microcanonical thermodynamics (MCTh) is contrasted to canonical thermodynamics (CTh). At phase transitions of 1.order the two ensembles are NOT equivalent even in the thermodynamic limit . Energy fluctuations do not vanish and phase separations are suppressed in CTh. A proper treatment of fluctuations is neccessary. MCTh allows to address even isolated small systems where phase transitions can be clearly classified into first order and continuous ones. The microcanonical caloric curve T(E) determines the transition temperature, latent heat AND surface entropy/tension. For systems of ca.1000 Na…
In the general case of a many-body Hamiltonian system, described by an autonomous Hamiltonian $H$, and with $K\geq 0$ independent conserved quantities, we derive the microcanonical thermodynamics. By a simple approach, based on the differential geometry, we derive the microcanonical entropy and the derivatives of the entropy with respect to the conserved quantities. In such a way, we show that all the thermodynamical quantities, as the temperature, the chemical potential or the specific heat, are measured as a microcanonical average of the appropriate microscopic dynamical functions that we ha…
The present study regards the zeroth order mean field approximation of a dipole-type interaction model, which is analytically solved in the canonical and microcanonical ensembles. After writing the canonical partition function, the free and internal energies, magnetization and the specific heat are derived and graphically represented. A crucial derivation is the calculation of the free energy, which is variationally evaluated, and it is shown that the exact result coincides with the approximate trend when $N$ tends to infinity. In the microcanonical ensemble, the entropy as other thermodynamic…
We determine the maximal work extractable via a cyclic Hamiltonian process from a positive-temperature ($T>0$) microcanonical state of a $N\gg 1$ spin bath. The work is much smaller than the total energy of the bath, but can be still much larger than the energy of a single bath spin, e.g. it can scale as ${\cal O}(\sqrt{N\ln N})$. Qualitatively same results are obtained for those cases, where the canonical state is unstable (e.g., due to a negative specific heat) and the microcanonical state is the only description of equilibrium. For a system coupled to a microcanonical bath the concept of fr…
We show that systems with negative specific heat can violate the zeroth law of thermodynamics. By both numerical simulations and by using exact expressions for free energy and microcanonical entropy it is shown that if two systems with the same intensive parameters but with negative specific heat are thermally coupled, they undergo a process in which the total entropy increases irreversibly. The final equilibrium is such that two phases appear, that is, the subsystems have different magnetizations and internal energies at temperatures which are equal in both systems, but that can be different …
The classical wave-particle Hamiltonian is considered in its generalized version, where two modes are assumed to interact with the co-evolving charged particles. The equilibrium statistical mechanics solution of the model is worked out analytically, both in the canonical and the microcanonical ensembles. The competition between the two modes is shown to yield ensemble inequivalence, at variance with the standard scenario where just one wave is allowed to develop. As a consequence, both temperature jumps and negative specific heat can show up in the microcanonical ensemble. The relevance of the…
Non-equilibrium systems under temperature modulation are investigated in the light of the stochastic thermodynamics. We show that, for small amplitudes of the temperature oscillations, the heat flux behaves sinusoidally with time, a result that allows the definition of the complex heat capacity. The real part of the complex heat capacity is the dynamic heat capacity, and its imaginary part is shown to be proportional to the rate of entropy production. We also show that the poles of the complex heat capacity are equal to imaginary unit multiplied by the eigenvalues of the unperturbed evolution …
We do Monte Carlo simulations of four 3D structural transitions, with vector-spin models of their martensitic strain domains under quenches to $T$, to test a generic post-quench Partial Equilibration Scenario (PES) of Ritort. We indeed confirm that energy-lowering passages between fixed-energy shells induce a signature PES distribution of an exponential tail in heat releases, scaled in an effective search temperature. A linear vanishing of this $T_{eff}(T)\sim T_d -T$ at a temperature $T_d$ where PES passage-searches freeze, explains the Vogel-Fulcher like divergence of equilibration times $ e…
We discuss Monte Carlo dynamics based on <N(sigma, Delta E)>_E, the (microcanonical) average number of potential moves which increase the energy by Delta E in a single spin flip. The microcanonical average can be sampled using Monte Carlo dynamics of a single spin flip with a transition rate min(1, <N(sigma', E-E')>_E' / <N(sigma, E'-E) >_E) from energy E to E'. A cumulative average (over Monte Carlo steps) can be used as a first approximation to the exact microcanonical average in the flip rate. The associated histogram is a constant independent of the energy. The canonical distribution of en…
In the study of 1D nonlinear Hamiltonian lattices, the conserved quantities play an important role in determining the actual behavior of heat conduction. Besides the total energy, total momentum and total stretch could also be conserved quantities. In microcanonical Hamiltonian dynamics, the total energy is always conserved. It was recently argued by Das and Dhar that whenever stretch (momentum) is not conserved in a 1D model, the momentum (stretch) and energy fields exhibit normal diffusion. In this work, we will systematically investigate the stretch diffusions for typical 1D nonlinear latti…
Seemingly different interpretations or descriptions of Einstein's model for the heat capacity of a solid can be found in textbooks and the literature. The purpose of this note is to clarify the equivalence of the different descriptions, which all lead to the same Einstein expression for the heat capacity of a solid of N atoms.
We study the global phase diagram of the infinite range Blume-Emery-Griffiths model both in the canonical and in the microcanonical ensembles. The canonical phase diagram is known to exhibit first order and continuous transition lines separated by a tricritical point. We find that below the tricritical point, when the canonical transition is first order, the phase diagrams of the two ensembles disagree. In this region the microcanonical ensemble exhibits energy ranges with negative specific heat and temperature jumps at transition energies. These results can be extended to weakly decaying noni…
We study the effect that randomness has on long-range interacting systems by using the ferromagnetic Ising model with $p$-body interactions in random fields. The case with p=2 yields a phase diagram similar to that of previously studied models and shows known features that inequivalence of the canonical and microcanonical ensembles brings with it, for example negative specific heat in a narrow region of the phase diagram. When p>2, however, the canonical phase diagram is completely different from the microcanonical one. The temperature does not necessarily determine the microcanonical phases u…
Development of steady state thermodynamics and statistical mechanics depends crucially on our ability to develop notions of equilibrium thermodynamics for nonequilibrium steady states (NESS). The present paper considers the development of heat capacity. A modified definition is proposed which continues to maintain the same relation to steady state Shannon entropy as in equilibrium, thus providing a thermodynamically consistent treatment of NESS heat capacity.
We explore two- and three-state Markov models driven out of thermal equilibrium by non-potential forces to demonstrate basic properties of the steady heat capacity based on the concept of quasistatic excess heat. It is shown that large enough driving forces can make the steady heat capacity negative. For both the low- and high-temperature regimes we propose an approximative thermodynamic scheme in terms of "dynamically renormalized" effective energy levels.
Thermal properties of few-fermion (n < 5) systems are investigated. The dependence of the heat capacity on the topology and shape of the cavity containing the particles is analyzed. It is found that the maximum of the heat capacity, occuring at low T, discussed recently by Toutounji for a system with n = 1 fermions, is even more visible for n = 2, but fades away for n = 3 and 4. For large T, the classical behavior is obtained; however, when T -> 0, the heat capacity tends to zero exponentially, not linearly, as in macroscopic and even mesoscopic systems. The physical relevance of these results…
According to thermodynamics, the specific heat of Boltzmannian short-range interacting systems is a positive quantity. Less intuitive properties are instead displayed by systems characterized by long-range interactions. In that case, the sign of specific heat depends on the considered statistical ensemble: negative specific heat can be found in isolated systems, which are studied in the framework of the microcanonical ensemble; on the other hand, it is generally recognized that a positive specific heat should always be measured in systems in contact with a thermal bath, for which the canonical…
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.
Using the stochastic thermodynamics, we determine the entropy production and the dynamic heat capacity of systems subject to a sinusoidally time dependent temperature, in which case the systems are permanently out of thermodynamic equilibrium inducing a continuous generation of entropy. The systems evolve in time according to a Fokker-Planck or to a Fokker-Planck-Kramers equation. Solutions of these equations, for the case of harmonic forces, are found exactly from which the heat flux, the production of entropy and the dynamic heat capacity are obtained as functions of the frequency of the tem…
Systems with long-range as well with short-range interactions should necessarily have a convex entropy S(E) at proper phase transitions of first order, i.e. when a separation of phases occurs. Here the microcanonical heat capacity c(E)= -\frac{(\partial S/\partial E)^2}{\partial^2S/\partial E^2} is negative. This should be observable even in macroscopic systems when energy fluctuations with the surrounding world can be sufficiently suppressed.
Calculating liquid energy and heat capacity in general form is an open problem in condensed matter physics. We develop a recent approach to liquids from the solid state by accounting for the contribution of anharmonicity and thermal expansion to liquid energy and heat capacity. We subsequently compare theoretical predictions to the experiments results of 5 commonly discussed liquids, and find a good agreement with no free fitting parameters. We discuss and compare the proposed theory to previous approaches.
The grand canonical ensemble lies at the core of quantum and classical statistical mechanics. A small system thermalizes to this ensemble while exchanging heat and particles with a bath. A quantum system may exchange quantities represented by operators that fail to commute. Whether such a system thermalizes and what form the thermal state has are questions about truly quantum thermodynamics. Here we investigate this thermal state from three perspectives. First, we introduce an approximate microcanonical ensemble. If this ensemble characterizes the system-and-bath composite, tracing out the bat…
We study non-canonical relaxation in a composite cold atoms system, consisting of subsystems that possess negative microcanonical specific heat. The system exhibits pre-thermalization far away from integrability due to the appearance of a single adiabatic invariant. The Thirring instability drives the constituent subsystems towards the edges of their allowed energy spectrum, thus greatly enhancing the contrast between the prethermal state and the long time thermal outcome.
