Nonequivalence Of Ensembles Research Articles

Ensemble nonequivalence and Bose–Einstein condensation in weighted networks

The asymptotic (non)equivalence of canonical and microcanonical ensembles, describing systems with soft and hard constraints respectively, is a central concept in statistical physics. Traditionally, the breakdown of ensemble equivalence (EE) has been associated with nonvanishing relative canonical fluctuations of the constraints in the thermodynamic limit. Recently, it has been reformulated in terms of a nonvanishing relative entropy density between microcanonical and canonical probabilities. The earliest observations of EE violation required phase transitions or long-range interactions. More recent research on binary networks found that an extensive number of local constraints can also break EE, even in absence of phase transitions. Here we study for the first time ensemble nonequivalence in weighted networks with local constraints. Unlike their binary counterparts, these networks can undergo a form of Bose–Einstein condensation (BEC) producing a core–periphery structure where a finite fraction of the link weights concentrates in the core. This phenomenon creates a unique setting where local constraints coexist with a phase transition. We find surviving relative fluctuations only in the condensed phase, as in more traditional BEC settings. However, we also find a non-vanishing relative entropy density for all temperatures, signalling a breakdown of EE due to the presence of an extensive number of constraints, irrespective of BEC. Therefore, in presence of extensively many local constraints, vanishing relative fluctuations no longer guarantee EE.

Strong ensemble nonequivalence in systems with local constraints

The asymptotic equivalence of canonical and microcanonical ensembles is a central concept in statistical physics, with important consequences for both theoretical research and practical applications. However, this property breaks down under certain circumstances. The most studied violation of ensemble equivalence requires phase transitions, in which case it has a ‘restricted’ (i.e. confined to a certain region in parameter space) but ‘strong’ (i.e. characterized by a difference between the entropies of the two ensembles that is of the same order as the entropies themselves) form. However, recent research on networks has shown that the presence of an extensive number of local constraints can lead to ensemble nonequivalence (EN) even in the absence of phase transitions. This occurs in a ‘weak’ (i.e. leading to a subleading entropy difference) but remarkably ‘unrestricted’ (i.e. valid in the entire parameter space) form. Here we look for more general manifestations of EN in arbitrary ensembles of matrices with given margins. These models have widespread applications in the study of spatially heterogeneous and/or temporally nonstationary systems, with consequences for the analysis of multivariate financial and neural time-series, multi-platform social activity, gene expression profiles and other big data. We confirm that EN appears in ‘unrestricted’ form throughout the entire parameter space due to the extensivity of local constraints. Surprisingly, at the same time it can also exhibit the ‘strong’ form. This novel, simultaneously ‘strong and unrestricted’ form of nonequivalence is very robust and imposes a principled choice of the ensemble. We calculate the proper mathematical quantities to be used in real-world applications.

First-order canonical phase transition as origin of the non-equivalence of the microcanonical and canonical ensembles in the thermodynamic limit

By using a simple argument, the cause of the thermodynamic non-equivalence of the canonical and microcanonical ensembles is clarified to be the first-order canonical phase transition in the thermodynamic limit. There is ensemble equivalence only when the energy fluctuation is negligible with respect to the average energy. The energy fluctuation does not vanish during the first-order transition even in the thermodynamic limit, so that the ensemble non-equivalence cannot hold anymore. Moreover, similar results can also be derived for other dual ensembles.

A condensed-matter analogue of the false vacuum

Through experimental investigation into the behaviour of a polar dielectric working fluid, an ideal ‘quasi-thermodynamic’ cycle has been established. Particular stages of this cycle are described in terms of a condensed-matter analogue of the false vacuum, when operating under negative pressure. The cycle is established between 37 °C and 15 °C under isochoric conditions. Phase-change work is created in two-directions, positive expansion-work and negative contraction-work. A large proportion of the expansion-work derives from a cooling process where the fluid exhibits negative heat capacity. When heat flux ceases, the fluid becomes unstable and heat capacity switches from negative to positive, displaying a ‘non-equivalence of ensembles’ phase-change. Whilst elements of the fluid behaviour can only be described by the statistical mechanics of non-equilibrium systems, the calculated equations of state for classical thermodynamics are confirmed to be accurate from the experimental investigation. However, the classical thermodynamic calculations for cycle-efficiency do not produce a symmetry of energy conservation. This suggests that an additional form of energy, having long-range interaction and distinct from heat and work input, is involved in the performance of the quasi-thermodynamic cycle. The expansion of a negative pressure fluid that contains inclusion compounds appears responsible for this potential energy interaction as an analogue of the false vacuum potential that can be explained by application of the virial theorem.

The statistical physics of real-world networks

In the last 15 years, statistical physics has been a very successful framework to model complex networks. On the theoretical side, this approach has brought novel insights into a variety of physical phenomena, such as self-organisation, scale invariance, emergence of mixed distributions and ensemble non-equivalence, that display unconventional features on heterogeneous networks. At the same time, thanks to their deep connection with information theory, statistical physics and the principle of maximum entropy have led to the definition of null models for networks reproducing some features of real-world systems, but otherwise as random as possible. We review here the statistical physics approach and the various null models for complex networks, focusing in particular on the analytic frameworks reproducing the local network features. We then show how these models have been used to detect statistically significant and predictive structural patterns in real-world networks, as well as to reconstruct the network structure in case of incomplete information. We further survey the statistical physics models that reproduce more complex, semi-local network features using Markov chain Monte Carlo sampling, as well as the models of generalised network structures such as multiplex networks, interacting networks and simplicial complexes.

Ensemble nonequivalence in random graphs with modular structure

Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must be met by every graph, soft constraints must be met only on average, subject to maximal entropy. In Squartini, de Mol, den Hollander and Garlaschelli (2015 New J. Phys. 17 023052) it was shown that ensembles of random graphs are nonequivalent when the degrees of the nodes are constrained, in the sense of a non-zero limiting specific relative entropy as the number of nodes diverges. In that paper, the nodes were placed either on a single layer (uni-partite graphs) or on two layers (bi-partite graphs). In the present paper we consider an arbitrary number of intra-connected and inter-connected layers, thus allowing for modular graphs with a multi-partite, multiplex, time-varying, block-model or community structure. We give a full classification of ensemble equivalence in the sparse regime, proving that breakdown occurs as soon as the number of local constraints (i.e. the number of constrained degrees) is extensive in the number of nodes, irrespective of the layer structure. In addition, we derive an explicit formula for the specific relative entropy and provide an interpretation of this formula in terms of Poissonisation of the degrees.

Breaking of Ensemble Equivalence in Networks.

It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. We find that, while graphs with a given number of links are ensemble equivalent, graphs with a given degree sequence are not. This result holds irrespective of whether the energy is nonadditive (as in unipartite graphs) or additive (as in bipartite graphs). In contrast with previous expectations, our results show that (1)physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by long-range interactions or nonadditivity, (2)mathematically, nonequivalence is determined by a different large-deviation behavior of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in general.

Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels

We present general and rigorous results showing that the microcanonical and canonical ensembles are equivalent at all three levels of description considered in statistical mechanics—namely, thermodynamics, equilibrium macrostates, and microstate measures—whenever the microcanonical entropy is concave as a function of the energy density in the thermodynamic limit. This is proved for any classical many-particle systems for which thermodynamic functions and equilibrium macrostates exist and are defined via large deviation principles, generalizing many previous results obtained for specific classes of systems and observables. Similar results hold for other dual ensembles, such as the canonical and grand-canonical ensembles, in addition to trajectory or path ensembles describing nonequilibrium systems driven in steady states.

A dynamical transition and metastability in a size-dependent zero-range process

We study a zero-range process with system-size dependent jump rates, which is known to exhibit a discontinuous condensation transition. Metastable homogeneous phases and condensed phases co-exist in extended phase regions around the transition, which have been fully characterized in the context of the equivalence and non-equivalence of ensembles. In this paper we report rigorous results on the large deviation properties and the free energy landscape which determine the metastable dynamics of the system. Within the condensed phase region we identify a new dynamic transition line which separates two distinct mechanisms of motion of the condensate, and provide a complete discussion of all relevant timescales. Our results are directly related to recent interest in metastable dynamics of condensing particle systems and apply to more general condensing particle systems, which exhibit the dynamical transition as a finite size effect.

Thermodynamic curvature and ensemble nonequivalence

In this work we consider thermodynamic geometries defined as Hessians of different potentials and derive some useful formulae that show their complementary role in the description of thermodynamic systems with two degrees of freedom that show ensemble nonequivalence. From the expressions derived for the metrics, we can obtain the curvature scalars in a very simple and compact form. We explain here the reason why each curvature scalar diverges over the line of divergence of one of the specific heats. This application is of special interest in the study of changes of stability in black holes as defined by Davies. From these results we are able to prove on a general footing a conjecture first formulated by Liu, L\"u, Luo and Shao stating that different Hessian metrics can correspond to different behaviors in the various ensembles. We study the case of two thermodynamic dimensions. Moreover, comparing our result with the more standard turning point method developed by Poincar\'e, we obtain that the divergence of the scalar curvature of the Hessian metric of one potential exactly matches the change of stability in the corresponding ensemble.

Effects of Thermal Fluctuations in the Interaction of a Small Number of Gas Particles with a Surface

The statistical mechanics of small systems is considered, where by virtue of physical or geometric limitations it is impossible to realize the transition to the thermodynamic limit. Fundamental effects are discussed which distinguish small equilibrium systems from large equilibrium systems: non-equivalence of ensembles, nonadditivity of thermodynamic potentials, the influence of fluctuations, complexity of the description of phase transitions. Along with a general analysis of features of the thermodynamic behavior of small systems, corresponding results are presented for the problem of adsorption of a gas on a surface.

Energy distribution and energy fluctuation in Tsallis statistics

The energy distribution and the energy fluctuation in the Tsallis canonical ensemble are studied with the OLM formalism but following a new way. The resulting formula for the energy fluctuation is not the same as that in previous work [L.Y. Liu, J.L. Du, Physica A 387 (2008) 5417]. In discussing the application of an ideal gas, we find that the energy fluctuation can not be negligible in the thermodynamic limit, showing the ensemble nonequivalence for this case in Tsallis statistics. We investigate the energy fluctuation with a Tsallis generalized canonical distribution studied by Plastino and Plastino [A.R. Plastino, A. Plastino, Phys. Lett. A 193 (1994) 140] for describing a system in contact with a finite heat bath. For this situation, the two formulae for the energy fluctuation are shown to be equivalent, while the nonextensive parameter q plays a very important role.

Nonequivalence of Ensembles for Long-Range Quantum Spin Systems in Optical Lattices

Motivated by the anisotropic long-range nature of the interactions between cold dipolar atoms or molecules in an optical lattice, we study the anisotropic quantum Heisenberg model with Curie-Weiss-type long-range interactions. Absence of a heat bath in optical lattice experiments suggests a study of this model within the microcanonical ensemble. The microcanonical entropy is calculated analytically and nonequivalence of microcanonical and canonical ensembles is found for a range of anisotropy parameters. From the shape of the entropy it follows that the Curie-Weiss Heisenberg model is indistinguishable from the Curie-Weiss Ising model in canonical thermodynamics, although their microcanonical thermodynamics differs. Qualitatively, the observed features of nonequivalent ensembles are expected to be relevant for long-range quantum spin systems realized in optical lattice experiments.

Discontinuous Condensation Transition and Nonequivalence of Ensembles in a Zero-Range Process

We study a zero-range process where the jump rates do not only depend on the local particle configuration, but also on the size of the system. Rigorous results on the equivalence of ensembles are presented, characterizing the occurrence of a condensation transition. In contrast to previous results, the phase transition is discontinuous and the system exhibits ergodicity breaking and metastable phases. This leads to a richer phase diagram, including nonequivalence of ensembles in certain phase regions. The paper is motivated by results from granular clustering, where these features have been observed experimentally.

Discontinuous Condensation Transition and Nonequivalence of Ensembles in a Zero-Range Process

We study a zero-range process where the jump rates do not only depend on the local particle configuration, but also on the size of the system. Rigorous results on the equivalence of ensembles are presented, characterizing the occurrence of a condensation transition. In contrast to previous results, the phase transition is discontinuous and the system exhibits ergodicity breaking and metastable phases. This leads to a richer phase diagram, including nonequivalence of ensembles in certain phase regions. The paper is motivated by results from granular clustering, where these features have been observed experimentally.

On the Mean-Field Spherical Model

Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical result allows for an exact discussion of the loci of the Fisher zeros of the canonical partition function. The microcanonical entropy is found to be nonanalytic for arbitrary finite N. The mean-field spherical model of finite size N is shown to be equivalent to a mixed isovector/isotensor sigma-model on a lattice of two sites. Partial equivalence of statistical ensembles is observed for the mean-field spherical model in the thermodynamic limit. A discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence.

Equilibrium states for the Bose gas

The generating functional of the cyclic representation of the canonical commutation relations (CCR) representation for the thermodynamic limit of the grand canonical ensemble of the free Bose gas with attractive boundary conditions is rigorously computed. We use it to study the condensate localization as a function of the homothety point for the thermodynamic limit using a sequence of growing convex containers. The Kac function is explicitly obtained proving nonequivalence of ensembles in the condensate region in spite of the condensate density being zero locally.

Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume–Emery–Griffiths model

We illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume–Emery–Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy. By computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the BEG model, we show that for values of the mean energy where the microcanonical entropy is nonconcave, the microcanonical distributions of states are nowhere realized in the canonical ensemble. Moreover, we show that for values of the mean energy where the microcanonical entropy is strictly concave, the equilibrium microcanonical distributions of states can be put in one-to-one correspondence with equivalent canonical equilibrium distributions of states. Our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by Ellis et al. (J. Stat. Phys. 101 (2000) 999).

Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows

Statistical equilibrium models of coherent structures intwo-dimensional and barotropic quasi-geostrophic turbulence areformulated using canonical and microcanonical ensembles, and theequivalence or nonequivalence of ensembles is investigated for thesemodels. The main results show that models in which the energy andcirculation invariants are treated microcanonically give richerfamilies of equilibria than models in which they are treatedcanonically. For each model, a variational principle thatcharacterizes its equilibrium states is derived by large deviationtechniques. An analysis of the two different variational principlesresulting from the canonical and microcanonical ensembles reveals thattheir equilibrium states coincide if and only if the microcanonicalentropy function is concave. Numerical computations implemented forgeostrophic turbulence over topography in a zonal channel demonstratethat nonequivalence of ensembles occurs over a wide range of the modelparameters and that physically interesting equilibria are oftenomitted by the canonical model. The nonlinear stability of the steadymean flows corresponding to microcanonical equilibria is establishedby a new Lyapunov argument. These stability theorems refine thewell-known Arnold stability theorems, which do not apply when themicrocanonical and canonical ensembles are not equivalent.

Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles

We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasigeostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions of an unconstrained minimization problem in which the constraint in the first problem is replaced by a Lagrange multiplier. The analysis of equivalence and nonequivalence of ensembles reduces to the following question in global optimization. What are the relationships between the set of solutions of the constrained minimization problem that characterizes microcanonical equilibrium macrostates and the set of solutions of the unconstrained minimization problem that characterizes canonical equilibrium macrostates? In general terms, our main result is that a necessary and sufficient condition for equivalence of ensembles to hold at the level of equilibrium macrostates is that it holds at the level of thermodynamic functions, which is the case if and only if the