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Abstract: Abstract This paper develops an averaging approach on macroscopic scales to derive Smoluchowski–Kramers approximation for a Langevin equation with state dependent friction in d-dimensional space. In this approach we couple the microscopic dynamics to the macroscopic scales. The weak convergence rate is also presented. PubDate: 2024-02-17

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Abstract: Abstract This paper deals with the problem of finding the Fokker Planck Equation (FPE) for the single-time probability density function (PDF) that optimally approximates the single-time PDF of a 1-D Stochastic Differential Equation (SDE) with Gaussian correlated noise. In this context, we tackle two main tasks. First, we consider the case of weak noise and in this framework we give a formal ground to the effective correction, introduced elsewhere (Bianucci and Mannella in J Phys Commun 4(10):105019, 2020, https://doi.org/10.1088/2399-6528/abc54e), to the Best Fokker Planck Equation (a standard “Born-Oppenheimer” result), also covering the more general cases of multiplicative SDE. Second, we consider the FPE obtained by using the Local Linearization Approach (LLA), and we show that a generalized cumulant approach allows an understanding of why the LLA FPE performs so well, even for noises with long (but finite) time scales and large intensities. PubDate: 2024-02-15

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Abstract: Abstract The dissipative spectral form factor (DSFF), recently introduced in Li et al. (Phys Rev Lett 127(17):170602, 2021) for the Ginibre ensemble, is a key tool to study universal properties of dissipative quantum systems. In this work we compute the DSFF for a large class of random matrices with real or complex entries up to an intermediate time scale, confirming the predictions from Li et al. (Phys Rev Lett 127(17):170602, 2021). The analytic formula for the DSFF in the real case was previously unknown. Furthermore, we show that for short times the connected component of the DSFF exhibits a non-universal correction depending on the fourth cumulant of the entries. These results are based on the central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices Cipolloni et al. (Electron J Prob 26:1–61, 2021) and Cipolloni et al. (Commun Pure Appl Math 76(5): 946–1034, 2023). PubDate: 2024-02-15

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Abstract: Abstract We study the radius of gyration \(R_T\) of a self-repellent fractional Brownian motion \(\left\{ B^H_t\right\} _{0\le t\le T}\) taking values in \(\mathbb {R}^d\) . Our sharpest result is for \(d=1\) , where we find that with high probability, $$\begin{aligned} R_T \asymp T^\nu , \quad \text {with }\quad \nu =\frac{2}{3}\left( 1+H\right) . \end{aligned}$$ For \(d>1\) , we provide upper and lower bounds for the exponent \(\nu \) , but these bounds do not match. PubDate: 2024-01-31

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Abstract: Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract theory, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems—independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models—without assuming detailed balance. For macroscopic equations arising out of these particle systems, we derive new variational formulations that generalise the classical gradient-flow formulation. PubDate: 2024-01-29

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Abstract: Abstract We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum —and, in particular, classical— interactions. Our approach is based on the use of “decoupling parameters”, advocated by Park (J. Stat. Phys. 27, 553–576 (1982)), which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a \(\beta \) -dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small \(\beta \) . To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams Bissacot et al. (J. Stat. Phys. 139, 598–617 (2010)). As an application we show that our lower bound of the radius of \(\beta \) -analyticity is larger than Park’s for quantum systems two-body interactions. PubDate: 2024-01-28

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Abstract: Abstract We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, Cannizzaro et al. (Ann Probab 50(6):2475–2498, 2022) proved sharp \(\sqrt{\log }\) -super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-D Gaussian Free Field (GFF) \(\underline{\omega }\) . We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of \(\underline{\omega }\) . Adapting their method, we show that if \(s\ge 1\) , with \(s=1\) corresponding to the standard stochastic heat equation, then the particle stays \(\sqrt{\log }\) -super diffusive, whereas if \(s<1\) , corresponding to a fractional heat equation, then the particle becomes diffusive. In fact, for \(s<1\) , we show that this is a particular case of Komorowski and Olla (J Funct Anal 197(1):179–211, 2003), which yields an invariance principle through a Sector Condition result. Our main results agree with the Alder–Wainwright scaling argument (see Alder and Wainwright in Phys Rev Lett 18:988–990, 1967; Alder and Wainwright in Phys Rev A 1:18–21, 1970; Alder et al. in Phys Rev A 4:233–237, 1971; Forster et al. in Phys Rev A 16:732–749, 1977) used originally in Tóth and Valkó (J Stat Phys 147(1):113–131, 2012) to predict the \(\log \) -corrections to diffusivity. We also provide examples which display \(\log ^a\) -super diffusive behaviour for \(a\in (0,1/2]\) . PubDate: 2024-01-28

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Abstract: Abstract This paper is concerned with the stochastic reaction-diffusion lattice systems defined on the entire set with both locally Lipschitz nonlinear drift and diffusion terms. The central limit theorem is derived for such infinite-dimensional stochastic systems. Moreover, the moderate deviation principle of solution processes is also established by the weak convergence method based on the variational representation of positive functionals of Brownian motion. The method relies on proving the convergence of the solutions of the controlled stochastic lattice systems. PubDate: 2024-01-28

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Abstract: Abstract In traffic flow, flyovers play a vital role in reducing traffic congestion, advancing commute timing, and helping to prevent collision scenarios. Inspired by the real-life applications of flyover, we model a coupled two-lane transport system where the lanes are divided into three segments in which the particle lane switching mechanism executes only in the middle segment of the lane, and particle inclusion and expulsion kinetics are performed in first and last segment of the lane to embody the infrastructure of flyover. The nature of the system characteristics is analyzed for diverse values of particle inclusion, expulsion, and coupling rates through graphical diagrams of phase planes, density profiles, shock dynamics, and phase transitions. The time-invariant behavior of the system is inspected numerically by utilizing the well-known method of mean-field theory, and it is discovered that the biased dynamics of particle inclusion, expulsion, and fully asymmetric coupling condition yield the phase diagram to disclose more unanticipated mixed stationary phases. Besides, the system experiences the fruitful novel incident of the double shock phase when the possibility of particle inclusion occurs relatively higher than the expulsion event. Also, it is noticed that the system experiences the fascinating phenomenon of shock propagation by traveling in either vertical/horizontal direction in the shock region of the phase diagram under the symmetric coupling circumstance, such as the position of the shock suddenly transits from the right to left segment without crossing the middle segment. The calculated numerical results are in good agreement with Monte Carlo simulation. PubDate: 2024-01-28

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Abstract: Abstract We present a method to compute transport coefficients in molecular dynamics. Transport coefficients quantify the linear dependencies of fluxes in non-equilibrium systems subject to small external forcings. Whereas standard non-equilibrium approaches fix the forcing and measure the average flux induced in the system driven out of equilibrium, a dual philosophy consists in fixing the value of the flux, and measuring the average magnitude of the forcing needed to induce it. A deterministic version of this approach, named Norton dynamics, was studied in the 1980s by Evans and Morriss. In this work, we introduce a stochastic version of this method, first developing a general formal theory for a broad class of diffusion processes, and then specializing it to underdamped Langevin dynamics, which are commonly used for molecular dynamics simulations. We provide numerical evidence that the stochastic Norton method provides an equivalent measure of the linear response, and in fact demonstrate that this equivalence extends well beyond the linear response regime. This work raises many intriguing questions, both from the theoretical and the numerical perspectives. PubDate: 2024-01-28

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Abstract: Abstract It is rigorously shown that an appropriate quantum annealing for any finite-dimensional spin system has no quantum first-order transition in transverse magnetization. This result can be applied to finite-dimensional spin-glass systems, where the ground state search problem is known to be hard to solve. Consequently, it is strongly suggested that the quantum first-order transition in transverse magnetization is not fatal to the difficulty of combinatorial optimization problems in quantum annealing. PubDate: 2024-01-28

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Abstract: Abstract We derive, within the replica formalism, a generalisation of the Crisanti–Sommers formula to describe the large deviation function (LDF) \(\mathcal{L}(e)\) for the speed-N atypical fluctuations of the intensive ground-state energy e of a generic spherical spin-glass in the presence of a random external magnetic field of variance \(\Gamma \) . We then analyse our exact formula for the LDF in much detail for the Replica symmetric, single step Replica Symmetry Breaking (1-RSB) and Full Replica Symmetry Breaking (FRSB) situations. Our main qualitative conclusion is that the level of RSB governing the LDF may be different from that for the typical ground-state. We find that while the deepest ground-states are always controlled by a LDF of replica symmetric form, beyond a finite threshold \(e\ge e_{t}\) a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the \(p=2\) spherical spin-glass (Fyodorov and Le Doussal in J Stat Phys 154:466, 2014) and the rigorous results by Dembo and Zeitouni (J Stat Phys 159:1306, 2015) which we are able to reproduce invoking an 1-RSB pattern. Finally at an even larger critical energy \(e_{c}\ge e_{t}\) , acting as a “wall”, the LDF diverges logarithmically, which we interpret as a change in the large deviation speed from N to a faster growth. In addition, we show that in the limit \(\Gamma \rightarrow 0\) the LDF takes non-trivial scaling forms (i) \(\mathcal{L}(e) \sim G((e-e_c)/\Gamma )\) in the vicinity of the wall (ii) \(\mathcal{L}(e) \sim \Gamma ^{\eta \nu } F((e-e_{\textrm{typ}})/\Gamma ^{\nu })\) in the vicinity of the typical energy, characterised by two new exponents \(\eta \ge 1\) and \(\nu \) characterising universality classes. Via matching the latter allows us to formulate several conjectures concerning the regime of typical fluctuations, identified as \(e-e_{\textrm{typ}} \sim N^{-1/\eta }\) and \(\Gamma \sim N^{-1/(\eta \nu )}\) . PubDate: 2024-01-27

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Abstract: Abstract We consider the one dimensional boundary driven harmonic model and its continuous version, both introduced in (Frassek et al. in J Stat Phys 180: 135–171, 2020). By combining duality and integrability the authors of (Frassek and Giardiná in J Math Phys 63: 103301, 2022) obtained the invariant measures in a combinatorial representation. Here we give an integral representation of the invariant measures which turns out to be a convex combination of inhomogeneous product of geometric distributions for the discrete model and a convex combination of inhomogeneous product of exponential distributions for the continuous one. The mean values of the geometric and of the exponential variables are distributed according to the order statistics of i.i.d. uniform random variables on a suitable interval fixed by the boundary sources. The result is obtained solving exactly the stationary condition written in terms of the joint generating function. The method has an interest in itself and can be generalized to study other models. We briefly discuss some applications. PubDate: 2024-01-20

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Abstract: Abstract From a unified vision of vector valued solutions in weighted Banach spaces, this paper establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unified approach for vector valued solutions in weighted Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities, angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate for p-binomial forms producing sharper estimates for the k-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only a \(2^+\) moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation. PubDate: 2024-01-17

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Abstract: Abstract This note establishes, first of all, the monotonic increase with N of the average K-body energy of classical N-body ground state configurations with \(N\ge K\) monomers that interact solely through a permutation-symmetric K-body potential, for any fixed integer \(K\ge 2\) . For the special case \(K=2\) this result had previously been proved, and used successfully as a test criterion for optimality of computer-generated lists of putative ground states of N-body clusters for various types of pairwise interactions. Second, related monotonicity results are established for N-monomer ground state configurations whose monomers interact through additive mixtures of certain types of k-meric potentials, \(k\in \{1,\ldots ,K\}\) , with \(K\ge 2\) fixed and \(N\ge K\) . All the monotonicity results furnish simple necessary conditions for optimality that any pertinent list of computer-generated putative global minimum energies for N-monomer clusters has to satisfy. As an application, databases of N-body cluster energies computed with an additive mix of the dimeric Lennard–Jones and trimeric Axilrod–Teller interactions are inspected. We also address how many local minima satisfy the upper bound inferred from the monotonicity conditions, both from a theoretical and from an empirical perspective. PubDate: 2024-01-04

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Abstract: Abstract We employ computer simulations to study the mechanical response and the associated structural transformations of porous glassy materials under cyclic pure shear deformation. The glassy samples are created via the rapid thermal quench and kinetically arrested solid gas phase separation in the athermal limit. Both the limit of high and low porosity systems are prepared by varying the average density. We consider two different strain amplitudes which correspond to near yielding and the steady state plastic flow regime. Under periodic loading, the system undergoes irreversible plastic rearrangements, leading to a gradual shift towards the lower potential energy minimum state and stress–strain hysteresis. The pore structure changes over consecutive cycles which are demonstrated in terms of pore size distribution function. With increasing shear cycle the distributions become skewed towards higher length scale as the adjacent pores coalesce and form larger pores. These results are found to be strongly dependent on the system density and the strain amplitude. Finally the evolution of the pore structure is studied by analyzing the average pore diameter with shear cycle. PubDate: 2024-01-03

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Abstract: Abstract We study the limit of a local average of the KPZ equation in dimension \(d=2\) with general initial data in the subcritical regime. Our result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equation and a Gaussian random variable that depends solely on the scale of averaging. This shows a unique mesoscopic averaging phenomenon that is only present in dimension two. Our work is inspired by the recent findings by Chatterjee (Ann l’Institut Henri Poincare (B) Probabilites et statistiques 59(2):774–794, 2023). PubDate: 2023-12-29

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Abstract: Abstract We replicate a renewal process at random times, which is equivalent to nesting two renewal processes, or considering a renewal process subject to stochastic resetting. We investigate the consequences on the statistical properties of the model of the intricate interplay between the two probability laws governing the distribution of time intervals between renewals, on the one hand, and of time intervals between resettings, on the other hand. In particular, the total number \({{\mathcal {N}}}_t\) of renewal events occurring within a specified observation time exhibits a remarkable range of behaviours, depending on the exponents characterising the power-law decays of the two probability distributions. Specifically, \({{\mathcal {N}}}_t\) can either grow linearly in time and have relatively negligible fluctuations, or grow subextensively over time while continuing to fluctuate. These behaviours highlight the dominance of the most regular process across all regions of the phase diagram. In the presence of Poissonian resetting, the statistics of \({{\mathcal {N}}}_t\) is described by a unique ‘dressed’ renewal process, which is a deformation of the renewal process without resetting. We also discuss the relevance of the present study to first passage under restart and to continuous time random walks subject to stochastic resetting. PubDate: 2023-12-28

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Abstract: Abstract We consider a one-dimensional spin model with the long-range random Hamiltonian given by \(H[\sigma ]=-\frac{1}{2}\sum _{x\ne y}\frac{ J_{x,y}\sigma _{x}\sigma _{y} }{ x-y ^{\alpha _{0}+\alpha _{x,y}}}\) . The randomness is considered in both the pairwise interaction \(J_{x,y}\) and in its decaying parameter with slowest value \(\alpha _{0}\) plus a non-negative random variable \(\alpha _{x,y}\) . We prove the loss of stability at \(\alpha _{0}=1/2\) . We also prove the existence of the free energy at the thermodynamic limit when \(\alpha _{0}>1/2\) . Furthermore, we show uniqueness of the equilibrium state for \(\alpha _{0}>3/2\) in the strong sense. PubDate: 2023-12-28

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Abstract: Abstract We investigate the probabilistic relevance in both sampling problems and combinatorics of trees of the Generalized Stirling numbers, as studied in Hsu and Shiue (Adv Appl Math 20(3):366–384, 1998). PubDate: 2023-12-28