Problems In Statistical Physics Research Articles

Asymptotic matching the self-consistent expansion to approximate the modified Bessel functions of the second kind

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem—too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified Bessel functions of the second kind, Kα(x) .

The Use of Computer Mathematics Systems for Solving Some Problems of Thermodynamics and Statistical Physics

With the help of the MathCAD mathematical package, solutions to problems using the heat balance equation were obtained, in which the heat capacity was calculated according to Debye theory, and spline interpolation of experimental data on heat capacity from the reference book was also used. The degree of helium ionization at a given temperature is calculated. An example of the influence of lithium impurity on the degree of hydrogen ionization is considered. The energy and pressure of neon in the region of multiple ionization are calculated. Isotherms of an ideal fermi gas are con structed. A graph of the Fermi-Dirac distribution function for various temperatures is constructed. The concentration of electrons and positrons at temperatures corresponding to the rest energy of electrons is calculated. These examples can be used in the educational process.

Spin glass theory and its new challenge: structured disorder

AbstractThis paper first describes, from a high-level viewpoint, the main challenges that had to be solved in order to develop a theory of spin glasses in the last fifty years. It then explains how important inference problems, notably those occurring in machine learning, can be formulated as problems in statistical physics of disordered systems. However, the main questions that we face in the analysis of deep networks require to develop a new chapter of spin glass theory, which will address the challenge of structured data.

Accelerating convergence of inference in the inverse Ising problem

Inferring modelling parameters of dynamical processes from observational data is an important inverse problem in statistical physics. In this paper, instead of passively observing the dynamics for inference, we focus on strategically manipulating dynamics to generate data that gives more accurate estimators within fewer observations. For this purpose, we consider the inference problem rooted in the Ising model with two opposite external fields, assuming that the strength distribution of one of the fields (labelled as passive) is unknown and needs to be inferred. In contrast, the other field (labelled active) is strategically deployed to interact with the Ising dynamics in such a way as to improve the accuracy of estimates of inferring the opposing passive field. By comparing to benchmark cases, we first demonstrate that it is possible to accelerate the inference by strategically interacting with the Ising dynamics. We then apply series expansions to obtain an approximation of the optimized influence configurations in the high-temperature region. Furthermore, by using mean-field estimates, we also demonstrate the applicability of the method in a more general scenario where real-time tracking of the system is infeasible. Last, analysing the optimized influence profiles, we describe heuristics for manipulating the Ising dynamics for faster inference. For example, we show that agents targeted more strongly by the passive field should also be strongly targeted by the active one.

Condensation of eigenmodes in functional brain network and its correlation to chimera state

Condensation has long been a closely studied problem in statistical physics but little attention has been paid to neural science. Here, we discuss this problem in brain networks and discover the condensation of a functional brain network whereby all its eigenmodes are condensed only into a few or even a single eigenmode of the structural brain network. We show that the condensation occurs due to the emergence of both chimera states and brain functions from the structure of the brain network. Furthermore, the condensation only appears in the regions of chimera and the condensed eigenmodes are only limited to the lower ones. Condensation is confirmed across different levels of brain subnetworks, including hemispheres, cognitive subnetworks, and isolated cognitive subnetworks, which are further supported by resting-state functional connectivity from empirical data. Our results indicate that condensation could be a potential mechanism for performing brain functions.

Control of Active Brownian Particles: An Exact Solution.

Control of stochastic systems is a challenging open problem in statistical physics, with a wealth of potential applications from biology to granulates. Unlike most cases investigated so far, we aim here at controlling a genuinely out-of-equilibrium system, the two dimensional active Brownian particles model in a harmonic potential, a paradigm for the study of self-propelled bacteria. We search for protocols for the driving parameters (stiffness of the potential and activity of the particles) bringing the system from an initial passivelike steady state to a final activelike one, within a chosen time interval. The exact analytical results found for this prototypical model of self-propelled particles brings control techniques to a wider class of out-of-equilibrium systems.

Estimation of small failure probabilities using the Accelerated Weight Histogram method

Simulation of rare events, such as small failure probabilities, is a common problem in several scientific and engineering fields. This paper presents a novel Monte-Carlo-based simulation approach for this purpose, called the Accelerated Weight Histogram (AWH) method. The method was originally developed to solve challenging sampling problems in statistical and biological physics, but its algorithm has here been reformulated for estimation of rare event probabilities. The applicability of the method is investigated for a couple of simpler computational examples and for a more advanced practical case, which consists of a rock tunnel stability problem. To estimate the probability of failure of the latter in a realistic manner, a boolean indicator function was used to describe failure, based on a mechanical concept known as unbalanced force ratio. The investigated cases indicate that the AWH method performs well for both simpler limit states and more complex failure definitions.

Emergent structural correlations in dense liquids.

The complete quantitative description of the structure of dense and supercooled liquids remains a notoriously difficult problem in statistical physics. Most studies to date focus solely on two-body structural correlations, and only a handful of papers have sought to consider additional three-body correlations. Here, we go beyond the state of the art by extracting many-body static structure factors from molecular dynamics simulations and by deriving accurate approximations up to the six-body structure factor via density functional theory. We find that supercooling manifestly increases four-body correlations, akin to the two- and three-body case. However, at small wave numbers, we observe that the four-point structure of a liquid drastically changes upon supercooling, both qualitatively and quantitatively, which is not the case in two-point structural correlations. This indicates that theories of the structure or dynamics of dense liquids should incorporate many-body correlations beyond the two-particle level to fully capture their intricate behavior.

Interface dynamics in the two-dimensional quantum Ising model

In a recent paper [Phys. Rev. Lett. 129, 120601] we have shown that the dynamics of interfaces, in the symmetry-broken phase of the two-dimensional ferromagnetic quantum Ising model, displays a robust form of ergodicity breaking. In this paper, we elaborate more on the issue. First, we discuss two classes of initial states on the square lattice, the dynamics of which is driven by complementary terms in the effective Hamiltonian and may be solved exactly: (a) strips of consecutive neighbouring spins aligned in the opposite direction of the surrounding spins, and (b) a large class of initial states, characterized by the presence of a well-defined "smooth" interface separating two infinitely extended regions with oppositely aligned spins. The evolution of the latter states can be mapped onto that of an effective one-dimensional fermionic chain, which is integrable in the infinite-coupling limit. In this case, deep connections with noteworthy results in mathematics emerge, as well as with similar problems in classical statistical physics. We present a detailed analysis of the evolution of these interfaces both on the lattice and in a suitable continuum limit, including the interface fluctuations and the dynamics of entanglement entropy. Second, we provide analytical and numerical evidence supporting the conclusion that the observed non-ergodicity -- arising from Stark localization of the effective fermionic excitations -- persists away from the infinite-Ising-coupling limit, and we highlight the presence of a timescale $T\sim e^{c L\ln L}$ for the decay of a region of large linear size $L$. The implications of our work for the classic problem of the decay of a false vacuum are also discussed.

Phononic thermal conduction and thermal regulation in low-dimensional micro-nano scale systems: Nonequilibrium statistical physics problems from chip heat dissipation

“Heat death”, namely, overheating, which will deteriorate the function of chips and eventually burn the device and has become an obstacle in the roadmap of the semiconductor industry. Therefore, heat dissipation becomes a key issue in further developing semiconductor. Heat conduction in chips encompasses the intricate dynamics of phonon conduction within one-dimensional, two-dimensional materials, as well as the intricate phonon transport through interfaces. In this paper, the research progress of the complexities of phonon transport on a nano and nanoscale in recent three years, especially the size dependent phonon thermal transport and the relationship between anomalous heat conduction and anomalous diffusion are summarized. Further discussed in this paper is the fundamental question within non-equilibrium statistical physics, particularly the necessary and sufficient condition for a given Hamiltonian whose macroscopic transport behavior obeys Fourier’s law. On the other hand, the methods of engineering the thermal conduction, encompassing nanophononic crystals, nanometamaterials, interfacial phenomena, and phonon condensation are also introduced. In order to comprehensively understand the phononic thermal conduction, a succinct overview of phonon heat transport phenomena, spanning from thermal quantization and the phonon Hall effect to the chiral phonons and their intricate interactions with other carriers is presented. Finally, the challenges and opportunities, and the potential application of phonons in quantum information are also discussed.

Lecture notes of the 15th international summer school on Fundamental Problems in Statistical Physics: Colloidal dispersions

In this brief contribution we review some basic concepts in the physics of soft matter focusing on colloidal dispersions. In the first part we discuss the phase behavior of hard colloidal particles and, building on a statistical mechanics formalism, we introduce the concept of effective interactions between colloids mediated by the presence of macromolecular additives in the dispersion. We then review some examples of effective interactions starting from the simplest entropy-driven mechanism observed in nature, i.e. the depletion effect, but also discussing more complex interactions as those mediated by a critical co-solute. In the second part, we introduce a novel type of colloids having particles “softness” as an extra control parameter. We focus on a particular type of soft colloid, called microgel, that has a polymeric nature since it is made of a cross-linked polymer network able to respond to external stimuli. Finally, to understand microgels responsiveness we discuss the thermodynamics of swelling of polymer networks described by the Flory-Rehner theory.

Speedup of the Metropolis protocol via algorithmic optimization

The Metropolis algorithm is widely used in Monte Carlo (MC) simulations in diverse areas of science and technology, especially for problems formulated in terms of lattice models. A common situation is the necessity to perform long sequential processing, e.g., when looking for equilibrium states of distinct physical systems. Hence, even marginal increases in efficiency of the algorithm individual steps can lead to significant reductions in absolute execution runtimes. Usual speedup procedures include hardware updates, parallelization (when possible) and sampling methods. Here we follow a different direction in trying to decrease the full execution times of MC approaches: algorithmic optimization. We show that the algorithms can be improved by implementing relatively few and simple changes in their organization and structure. First, we discuss some refinements for the pseudo-random number generator, addressing the broadly employed Mersenne-Twister algorithm (MT19937-64). Second, we develop a protocol to precalculate the Boltzmann factor, thereby avoiding the high cost of repeatedly calls to this exponential function (indeed, a very recurring step in the standard Metropolis method). To benchmark our proposals we choose the Ising model since it is one of the best known and more extensively studied problems in statistical physics. We consider the mentioned optimizations and different computational elements, like compilers and Hamiltonian variables ranges, testing the efficiency to obtain the system solutions. Our results suggest that the present set of improvement schemes—namely; decreasing the processing time for both, to generate a random number and to implement the one-flip Metropolis step; systematically enforcing optimization for the maximum quantity of algorithm structures accessing random numbers in a code; and considerably reducing the amount of required computations of the MC probabilistic actualization term—might constitute a relevant addition to the existing collection of expediting techniques in MC computational routines.

Floquet States in Open Quantum Systems

In Floquet engineering, periodic driving is used to realize novel phases of matter that are inaccessible in thermal equilibrium. For this purpose, the Floquet theory provides us a recipe for obtaining a static effective Hamiltonian. Although many existing works have treated closed systems, it is important to consider the effect of dissipation, which is ubiquitous in nature. Understanding the interplay of periodic driving and dissipation is a fundamental problem of nonequilibrium statistical physics that is receiving growing interest because of the fact that experimental advances have allowed us to engineer dissipation in a controllable manner. In this review, we give a detailed exposition on the formalism of quantum master equations for open Floquet systems and highlight recent work investigating whether equilibrium statistical mechanics applies to Floquet states.

Emergent second law for non-equilibrium steady states

The Gibbs distribution universally characterizes states of thermal equilibrium. In order to extend the Gibbs distribution to non-equilibrium steady states, one must relate the self-information {{{{{{{mathcal{I}}}}}}}}(x)=-!log ({P}_{{{{{{{{rm{ss}}}}}}}}}(x)) of microstate x to measurable physical quantities. This is a central problem in non-equilibrium statistical physics. By considering open systems described by stochastic dynamics which become deterministic in the macroscopic limit, we show that changes {{Delta }}{{{{{{{mathcal{I}}}}}}}}={{{{{{{mathcal{I}}}}}}}}({x}_{t})-{{{{{{{mathcal{I}}}}}}}}({x}_{0}) in steady state self-information along deterministic trajectories can be bounded by the macroscopic entropy production Σ. This bound takes the form of an emergent second law {{Sigma }}+{k}_{b}{{Delta }}{{{{{{{mathcal{I}}}}}}}},ge ,0, which contains the usual second law Σ ≥ 0 as a corollary, and is saturated in the linear regime close to equilibrium. We thus obtain a tighter version of the second law of thermodynamics that provides a link between the deterministic relaxation of a system and the non-equilibrium fluctuations at steady state. In addition to its fundamental value, our result leads to novel methods for computing non-equilibrium distributions, providing a deterministic alternative to Gillespie simulations or spectral methods.

Statistics of diffusive encounters with a small target: three complementary approaches

Diffusive search for a static target is a common problem in statistical physics with numerous applications in chemistry and biology. We look at this problem from a different perspective and investigate the statistics of encounters between the diffusing particle and the target. While an exact solution of this problem was recently derived in the form of a spectral expansion over the eigenbasis of the Dirichlet-to-Neumann operator, the latter is generally difficult to access for an arbitrary target. In this paper, we present three complementary approaches to approximate the probability density of the rescaled number of encounters with a small target in a bounded confining domain. In particular, we derive a simple fully explicit approximation, which depends only on a few geometric characteristics such as the surface area and the harmonic capacity of the target, and the volume of the confining domain. We discuss the advantages and limitations of three approaches and check their accuracy. We also deduce an explicit approximation for the distribution of the first-crossing time, at which the number of encounters exceeds a prescribed threshold. Its relations to common first-passage time problems are discussed.

Mean exit time in irregularly-shaped annular and composite disc domains

Calculating the mean exit time (MET) for models of diffusion is a classical problem in statistical physics, with various applications in biophysics, economics and heat and mass transfer. While many exact results for MET are known for diffusion in simple geometries involving homogeneous materials, calculating MET for diffusion in realistic geometries involving heterogeneous materials is typically limited to repeated stochastic simulations or numerical solutions of the associated boundary value problem (BVP). In this work we derive exact solutions for the MET in irregular annular domains, including some applications where diffusion occurs in heterogenous media. These solutions are obtained by taking the exact results for MET in an annulus, and then constructing various perturbation solutions to account for the irregular geometries involved. These solutions, with a range of boundary conditions, are implemented symbolically and compare very well with averaged data from repeated stochastic simulations and with numerical solutions of the associated BVP. Software to implement the exact solutions is available on GitHub.

Lower bounds for Ramsey numbers as a statistical physics problem

Ramsey’s theorem, concerning the guarantee of certain monochromatic patterns in large enough edge-coloured complete graphs, is a fundamental result in combinatorial mathematics. In this work, we highlight the connection between this abstract setting and a statistical physics problem. Specifically, we design a classical Hamiltonian that favours configurations in a way to establish lower bounds on Ramsey numbers. As a proof of principle we then use Monte Carlo methods to obtain such lower bounds, finding rough agreement with known literature values in a few cases we investigated. We discuss numerical limitations of our approach and indicate a path towards the treatment of larger graph sizes.

Heating Rates under Fast Periodic Driving beyond Linear Response.

Heating under periodic driving is a generic nonequilibrium phenomenon, and it is a challenging problem in nonequilibrium statistical physics to derive a quantitatively accurate heating rate. In this work, we provide a simple formula on the heating rate under fast and strong periodic driving in classical and quantum many-body systems. The key idea behind the formula is constructing a time-dependent dressed Hamiltonian by moving to a rotating frame, which is found by a truncation of the high-frequency expansion of the micromotion operator, and applying the linear-response theory. It is confirmed for specific classical and quantum models that the second-order truncation of the high-frequency expansion yields quantitatively accurate heating rates beyond the linear-response regime. Our result implies that the information on heating dynamics is encoded in the first few terms of the high-frequency expansion, although heating is often associated with an asymptotically divergent behavior of the high-frequency expansion.

Series Expansion of the Excess Work Using Nonlinear Response Theory

The calculation of observable averages in non-equilibrium regimes is one of the most important problems in statistical physics. Using the Hamiltonian approach of nonlinear response theory, we obtain a series expansion of the average excess work and illustrate it with specific examples of thermally isolated systems. We report the emergence of non-vanishing contributions for large switching times when the system is subjected to strong driving. The problem is solved by using an adapted multiple-scale method to suppress these secular terms. Our paradigmatic examples show how the method is implemented generating a truncated series that obeys the Second Law of Thermodynamics.

Turbulent plane Poiseuille flow

The understanding of fully developed turbulence remains a major unsolved problem of statistical physics. A challenge there is how to use what we understand of this problem to build-up a closure method, that is to express the time-averaged turbulent stress tensor as a function of the time-averaged velocity field u(x). We have shown that the closure problem is strongly restricted due to constraints on the time-averaged quantities, and to scaling laws derived from the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations. It implies that the turbulent stress is a non-local function in space of the time-averaged velocity u(x), involving an integral kernel, an extension of classical Boussinesq theory of turbulent viscosity. We treat one of the simplest possible physical situation, the turbulent Poiseuille flow between two parallel plates. In this case, the integral kernel takes a simple form leading to full analysis of the time-averaged turbulent flow. In the limit of a very large Reynolds number, one has to match a viscous boundary layer near the walls bounding the flow and an outer solution in the bulk of the flow, a non-trivial asymptotic analysis because of logarithms. Besides the boundary layers close to the walls, there is another “inner” boundary layer near the center plane of the flow. Our expression for the turbulent stress tensor yields ultimately the complete structure of the boundary layer, including in locations where viscosity becomes important.