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. An erratum to this article can be found at     

Condensation in the Inclusion Process and Related Models

We study condensation in several particle systems related to the inclusion process. For an asymmetric one-dimensional version with closed boundary conditions and drift to the right, we show that all but a finite number of particles condense on the right-most site. This is extended to a general result for independent random variables with different tails, where condensation occurs for the index (site) with the heaviest tail, generalizing also previous results for zero-range processes. For inclusion processes with homogeneous stationary measures we establish condensation in the limit of vanishing diffusion strength in the dynamics, and give several details about how the limit is approached for finite and infinite systems. Finally, we consider a continuous model dual to the inclusion process, the so-called Brownian energy process, and prove similar condensation results.     

The Equilibrium States for a Model with Two Kinds of Bose Condensation

We study the equilibrium Gibbs states for a Boson gas model, defined by Bru and Zagrebnov, which has two phase transitions of the Bose condensation type. The two phase transitions correspond to two distinct mechanisms by which these condensations can occur. The first (non-conventional) Bose condensation is mediated by a zero-mode interaction term in the Hamiltonian. The second is a transition due to saturation quite similar to the conventional Bose–Einstein (BE) condensation in the ideal Bose gas. Due to repulsive interaction in non-zero modes the model manifests a generalized type III; i.e., non-extensive BE condensation. Our main result is that, as in the ideal Bose gas, the conventional condensation is accompanied by a loss of strong equivalence of the canonical and grand canonical ensembles whereas the non-conventional one, due to the interaction, does not break the equivalence of ensembles, at least not on the level of the gauge invariant states. It is also interesting to note that the type of (generalized) condensate, I, II, or III (in the terminology of van den Berg, Lewis, and Pulé), has no effect on the equivalence of ensembles. These results are proved by computing the generating functional of the cyclic representation of the Canonical Commutation Relation (CCR) for the corresponding equilibrium Gibbs states.     

General linear response formula for non integrable systems obeying the Vlasov equation

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.     

Zero-Range Process with Open Boundaries

We calculate the exact stationary distribution of the one-dimensional zero-range process with open boundaries for arbitrary bulk and boundary hopping rates. When such a distribution exists, the steady state has no correlations between sites and is uniquely characterized by a space-dependent fugacity which is a function of the boundary rates and the hopping asymmetry. For strong boundary drive the system has no stationary distribution. In systems which on a ring geometry allow for a condensation transition, a condensate develops at one or both boundary sites. On all other sites the particle distribution approaches a product measure with the finite critical density ρ_c. In systems which do not support condensation on a ring, strong boundary drive leads to a condensate at the boundary. However, in this case the local particle density in the interior exhibits a complex algebraic growth in time. We calculate the bulk and boundary growth exponents as a function of the system parameters.     

No Current Without Heat

We show for a large class of interacting particle systems that whenever the stationary measure is not reversible for the dynamics, then the mean entropy production in the steady state is strictly positive. This extends to the thermodynamic limit the equivalence between microscopic reversibility and zero mean entropy production: time-reversal invariance cannot be spontaneously broken.     

Gibbsian Stationary Non-equilibrium States

We study the structure of stationary non-equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. Since divergence free flows are characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. According to this, for example, the instantaneous current of any interacting particle system on a finite torus can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components are associated with functions on the configuration space. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field and we use this decomposition to construct models having a fixed invariant measure.     

Equivalence of Ensembles Under Inhomogeneous Conditioning and Its Applications to Random Young Diagrams

We prove the equivalence of ensembles or a realization of the local equilibrium for Bernoulli measures on ${\mathbb{Z}}$ conditioned on two conserved quantities under the situation that one of them is spatially inhomogeneous. For the proof, we extend the classical local limit theorem for a sum of Bernoulli independent sequences to those multiplied by linearly growing weights. The motivation comes from the study of random Young diagrams and their evolutional models, which were originally suggested by Herbert Spohn. We discuss the relation between our result and the so-called Vershik curve which appears in a scaling limit for height functions of two-dimensional Young diagrams. We also discuss a related random dynamics.     

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, quasi-geostrophic 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 microcanonical entropy is concave. The necessity of this condition is new and has the following striking formulation. If the microcanonical entropy is not concave at some value of its argument, then the ensembles are nonequivalent in the sense that the corresponding set of microcanonical equilibrium macrostates is disjoint from any set of canonical equilibrium macrostates. We point out a number of models of physical interest in which nonconcave microcanonical entropies arise. We also introduce a new class of ensembles called mixed ensembles, obtained by treating a subset of the dynamical invariants canonically and the complementary set microcanonically. Such ensembles arise naturally in applications where there are several independent dynamical invariants, including models of dispersive waves for the nonlinear Schrödinger equation. Complete equivalence and nonequivalence results are presented at the level of equilibrium macrostates for the pure canonical, the pure microcanonical, and the mixed ensembles.     

Relativistic non-Hamiltonian mechanics

Relativistic particle subjected to a general four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_μu^μ + c² = 0, where c is the speed of light in vacuum. In the general case, four-forces are non-potential, and the relativistic particle is a non-Hamiltonian system in four-dimensional pseudo-Euclidean space-time. We consider non-Hamiltonian and dissipative systems in relativistic mechanics. Covariant forms of the principle of stationary action and the Hamilton’s principle for relativistic mechanics of non-Hamiltonian systems are discussed. The equivalence of these principles is considered for relativistic particles subjected to potential and non-potential forces. We note that the equations of motion which follow from the Hamilton’s principle are not equivalent to the equations which follow from the variational principle of stationary action. The Hamilton’s principle and the principle of stationary action are not compatible in the case of systems with nonholonomic constraint and the potential forces. The principle of stationary action for relativistic particle subjected to non-potential forces can be used if the Helmholtz conditions are satisfied. The Hamilton’s principle and the principle of stationary action are equivalent only for a special class of relativistic non-Hamiltonian systems.     

The Cut Metric,Random Graphs,and Branching Processes

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).     

Density-functional theory of inhomogeneous fluids in the canonical ensemble

We present a density-functional approach for dealing with inhomogeneous fluids in the canonical ensemble. A general relation is proposed between the free-energy functionals in the canonical and the grand canonical ensembles. The minimization of the canonical-ensemble free-energy functional gives rise to Euler-Lagrange equations which involve averaged Ornstein-Zernike equations of second and third order. The theory is especially appropriate for systems with a small, fixed number of particles. As an example of application we obtain accurate results for the density profile of a hard-sphere fluid in a closed spherical cavity that contains only a few particles.     

Non-equivalence of Dynamical Ensembles and Emergent Non-ergodicity

Dynamical ensembles have been introduced to study constrained stochastic processes. In the microcanonical ensemble, the value of a dynamical observable is constrained to a given value. In the canonical ensemble a bias is introduced in the process to move the mean value of this observable. The equivalence between the two ensembles means that calculations in one or the other ensemble lead to the same result. In this paper, we study the physical conditions associated with ensemble equivalence and the consequences of non-equivalence. For continuous time Markov jump processes, we show that ergodicity guarantees ensemble equivalence. For non-ergodic systems or systems with emergent ergodicity breaking, we adapt a method developed for equilibrium ensembles to compute asymptotic probabilities while caring about the initial condition. We illustrate our results on the infinite range Ising model by characterizing the fluctuations of magnetization and activity. We discuss the emergence of non-ergodicity by showing that the initial condition can only be forgotten after a time that scales exponentially with the number of spins.

     

Stationary Nonequilibrium Particle-Number Distribution Density Formed by Particle Sources and Sinks

We analyze an inhomogeneous diffusion equation with the given potential profile and the particle source and sink. The possibility of forming a nonequilibrium but stationary state of the system is discussed. For a time-constant point source, we obtain general formulas determining the nonequilibrium stationary density distributions of the particle number as a function of potential-profile shape. The influence of sources and sinks on the flow structure in a diffusion system is determined. Some examples are considered.     

Phase transition of trapped interacting Bose gases

We review some recent theoretical work on the phase transition of interacting Bose gases in the presence of external trapping potentials. A general framework for the study of such questions is presented which is based on the application of perturbative momentum-shell renormalization group methods to the trapped gas in the uncondensed phase. After giving an overview of this approach, we first establish its validity by comparing to previous results for homogeneous and harmonically trapped gases. Using this theoretical framework, we then examine various aspects of how external potentials influence the physics of condensation. (i) By studying the case of general power-law potentials and complemented by arguments from variational perturbation theory, it is quantitatively worked out how a growing inhomogeneity of the trapping potential diminishes nonperturbative effects at the transition. (ii) It is shown how by superimposing a weak periodic potentials on the homogeneous system, the characteristic nonperturbative momentum scale of critical interacting Bose gases can be probed. (iii) For a gas in a random potential, it is studied how condensation is affected by the combined influence of disorder effects and particle interactions.     

Phase coexistence in driven one-dimensional transport

We study a one-dimensional totally asymmetric exclusion process with random particle attachments and detachments in the bulk. The resulting dynamics leads to unexpected stationary regimes for large but finite systems. Such regimes are characterized by a phase coexistence of low and high density regions separated by domain walls. We use a mean-field approach to interpret the numerical results obtained by Monte Carlo simulations, and we predict the phase diagram of this nonconserved dynamics in the thermodynamic limit.     

Multipole Radiation in a Collisionless Gas Coupled to Electromagnetism or Scalar Gravitation

We consider the relativistic Vlasov-Maxwell and Vlasov-Nordström systems which describe large particle ensembles interacting by either electromagnetic fields or a relativistic scalar gravity model. For both systems we derive a radiation formula analogous to the Einstein quadrupole formula in general relativity.Supported in parts by DFG priority research program SPP 1095     

Embedded random matrix ensembles for complexity and chaos in finite interacting particle systems

Universal properties of simple quantum systems whose classical counter parts are chaotic, are modeled by the classical random matrix ensembles and their interpolations/deformations. However for finite interacting many-particle systems such as atoms, molecules, nuclei and mesoscopic systems (atomic clusters, helium droplets, quantum dots, etc.) for wider range of phenomena, it is essential to include information such as particle number, number of single-particle orbits, lower particle rank of the interaction, etc. These considerations led to resurgence of interest in investigating in detail the so-called embedded random matrix ensembles and their various deformed versions. Besides giving a overview of the basic results of embedded ensembles for the smoothed state densities and transition matrix elements, recent progress in investigating these ensembles with various deformations, for deriving a statistical mechanics (with relationships between quantum chaos, thermalization, phase transitions and Fock space localization, etc.) for isolated finite systems with few particles is briefly discussed. These results constitute new progress in deriving a basis for statistical spectroscopy (introduced and applied in nuclear structure physics and more recently in atomic physics) and its domains of applicability.     

Energy spectrum of a nonstationary ensemble of pulses

We introduce a new definition of the energy spectrum of a nonstationary ensemble of pulses that reduces to the usual ones in the limit of statistically stationary ensembles of signals and of fully temporarily coherent ensembles.