Journal of Statistical Physics最新文献

We define infinite tensor product spaces that extend Fock space, and allow for implementing Bogoliubov transformations which violate the Shale or Shale–Stinespring condition. So an implementation on the usual Fock space would not be possible. Both the bosonic and fermionic case are covered. Conditions for implementability in an extended sense are stated and proved. From these, we derive conditions for a quadratic Hamiltonian to be diagonalizable by a Bogoliubov transformation that is implementable in the extended sense. We apply our results to Bogoliubov transformations from quadratic bosonic interactions and BCS models, where the Shale or Shale–Stinespring condition is violated, but an extended implementation nevertheless works.

I introduce a lattice gas model with excluded volume interactions and investigate its phase behavior on the square lattice. Monte Carlo simulations and finite-size scaling show a discontinuous sub-lattice ordering transition towards a high-density solidlike phase characterised by an anti-ferromagnetic dimerized ground state with eightfold degeneracy. Since thermodynamics is controlled by entropy alone this provides a simple athermal realization of a dimerization process. Interestingly, the model can be considered as a special case of a family of hard-core lattice gases with local geometric constraints on both particles and vacancies, which generalizes the one proposed by Biroli and Mézard in the context of glassy systems.

We consider reversible ergodic Markov chains with finite state space, and we introduce a new notion of quasi-stationary distribution that does not require the presence of any absorbing state. In our setting, the hitting time of the absorbing set is replaced by an optimal strong stationary time, representing the “hitting time of the stationary distribution”. On the one hand, we show that our notion of quasi-stationary distribution corresponds to the natural generalization of the Yaglom limit. On the other hand, similarly to the classical quasi-stationary distribution, we show that it can be written in terms of the eigenvectors of the underlying Markov kernel, and it is therefore amenable of a geometric interpretation. Moreover, we recover the usual exponential behavior that characterizes quasi-stationary distributions and metastable systems. We also provide some toy examples, which show that the phenomenology is richer compared to the absorbing case. Finally, we present some counterexamples, showing that the assumption on the reversibility cannot be weakened in general.

We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the Bernard-LeClair scheme to the arena of classical stochastic processes and leading to a set of up to fifteen allowed symmetry classes. We construct families of solutions of arbitrary matrix dimensions for five of these classes with a simple physical interpretation of particles hopping on multipartite graphs. In the remaining classes, such a simple construction is prevented by the positivity of entries of the generator particular to classical stochastic processes, which imposes a further requirement beyond the usual symmetry classification constraints. We partially overcome this difficulty by resorting to a stochastic optimization algorithm, finding specific examples of generators of small matrix dimensions in six further classes, leaving the existence of the final four allowed classes an open problem. Our symmetry-based results unveil new possibilities in the dynamics of classical stochastic processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the spectra of Markov generators, and time reversal properties of stochastic trajectories and correlation functions.

A common and effective method for calculating the steady-state distribution of a process under stochastic resetting is the renewal approach that requires only the knowledge of the reset-free propagator of the underlying process and the resetting time distribution. The renewal approach is widely used for simple model systems such as a freely diffusing particle with exponentially distributed resetting times. However, in many real-world physical systems, the propagator, the resetting time distribution, or both are not always known beforehand. In this study, we develop a numerical renewal method to determine the steady-state probability distribution of particle positions based on the measured system propagator in the absence of resetting combined with the known or measured resetting time distribution. We apply and validate our method in two distinct systems: one involving interacting particles and the other featuring strong environmental memory. Thus, the renewal approach can be used to predict the steady state under stochastic resetting of any system, provided that the free propagator can be measured and that it undergoes complete resetting.

We prove that Villain interaction applied to lattice gauge theory can be obtained as the limit of both Wilson and Manton interactions on a larger graph which we call the carpet graph. This is the lattice gauge theory analog of a well-known property for spin O(N) models where Villain type interactions are the limit of (mathbb {S}^{N-1}) spin systems defined on a cable graph. Perhaps surprisingly in the setting of lattice gauge theory, our proof also applies to non-Abelian lattice theory such as SU(3)-lattice gauge theory and its limiting Villain interaction. In the particular case of an Abelian lattice gauge theory, this allows us to extend the validity of Ginibre inequality to the case of the Villain interaction.

We investigate a class of stochastic chemical reaction networks with (n{ge }1) chemical species (S_1), ..., (S_n), and whose complexes are only of the form (k_iS_i), (i{=}1),..., n, where ((k_i)) are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter N. A natural hierarchy of fast processes, a subset of the coordinates of ((X_i(t))), is determined by the values of the mapping (i{mapsto }k_i). We show that the scaled vector of coordinates i such that (k_i{=}1) and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as N gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.

In this paper, we study the stability and bifurcations of spontaneous stochasticity using an approach reminiscent of the Feigenbaum renormalization group (RG). We consider dynamical models on a self-similar space-time lattice as toy models for multiscale motion in hydrodynamic turbulence. Here an ill-posed ideal system is regularized at small scales and the vanishing regularization (inviscid) limit is considered. By relating the inviscid limit to the dynamics of the RG operator acting on the flow maps, we explain the existence and universality (regularization independence) of the limiting solutions as a consequence of the fixed-point RG attractor. Considering the local linearized dynamics, we show that the convergence to the inviscid limit is governed by the universal RG eigenmode. We also demonstrate that the RG attractor undergoes a period-doubling bifurcation with parameter variation, thereby changing the nature of the inviscid limit. In the case of chaotic RG dynamics, we introduce the stochastic RG operator acting on Markov kernels. Then the RG attractor becomes stochastic, which explains the existence and universality of spontaneously stochastic solutions in the limit of vanishing noise. We study a linearized structure (RG eigenmode) of the stochastic RG attractor and its period-doubling bifurcation. Viewed as prototypes of Eulerian spontaneous stochasticity, our models explain its mechanism, universality and potential diversity.

The continuous random energy model (CREM) is a toy model of disordered systems introduced by Bovier and Kurkova in 2004 based on previous work by Derrida and Spohn in the 80s. In a recent paper by Addario-Berry and Maillard, they raised the following question: what is the threshold (beta _G), at which sampling approximately the Gibbs measure at any inverse temperature (beta >beta _G) becomes algorithmically hard? Here, sampling approximately means that the Kullback–Leibler divergence from the output law of the algorithm to the Gibbs measure is of order o(N) with probability approaching 1, as (Nrightarrow infty ), and algorithmically hard means that the running time, the numbers of vertices queries by the algorithms, is beyond of polynomial order. The present work shows that when the covariance function A of the CREM is concave, for all (beta >0), a recursive sampling algorithm on a renormalized tree approximates the Gibbs measure with running time of order (O(N^{1+varepsilon })). For A non-concave, the present work exhibits a threshold (beta _G<infty ) such that the following hardness transition occurs: (a) For every (beta le beta _G), the recursive sampling algorithm approximates the Gibbs measure with a running time of order (O(N^{1+varepsilon })). (b) For every (beta >beta _G), a hardness result is established for a large class of algorithms. Namely, for any algorithm from this class that samples the Gibbs measure approximately, there exists (z>0) such that the running time of this algorithm is at least (e^{zN}) with probability approaching 1. In other words, it is impossible to sample approximately in polynomial-time the Gibbs measure in this regime. Additionally, we provide a lower bound of the free energy of the CREM that could hold its value.

In this work we investigate a 1D evolution equation involving a divergence form operator where the diffusion coefficient inside the divergence changes sign, as in models for metamaterials. We focus on the construction of a fundamental solution for the evolution equation, which does not proceed as in the case of standard parabolic PDE’s, since the associated second order operator is not elliptic. We show that a spectral representation of the semigroup associated to the equation can be derived, which leads to a first expression of the fundamental solution. We also derive a probabilistic representation in terms of a pseudo Skew Brownian Motion (SBM). This construction generalizes that derived from the killed SBM when the diffusion coefficient is piecewise constant but remains positive. We show that the pseudo SBM can be approached by a rescaled pseudo asymmetric random walk, which allows us to derive several numerical schemes for the resolution of the PDE and we report the associated numerical test results.