Statistical Mechanics Perspective Research Articles

New insights into the prediction for the potential of soil organic carbon accumulation: From the perspective of non-equilibrium statistical mechanics

The accumulation of soil organic carbon (SOC) is significant for soil health and ecosystem services. Numerous studies have assessed the dynamic changes of SOC by considering the microbial system as an equilibrium system. However, they failed to reveal the complexity of the SOC accumulation/loss process, as the microbial system is a non-equilibrium system affected by stochastic fluctuations from the external environment. This study is the first to explore the complex non-equilibrium relationship between microbial carbon use efficiency (CUE) and SOC by using potential landscape and flux in non-equilibrium statistical mechanics. Nitrogen (N) was identified as the most critical environmental factor influencing CUE on a global scale, with the transition between the carbon loss state and the carbon sequestration state observed along N gradients. Random perturbations of other environmental factors could also trigger transition. Non-equilibrium thermodynamic quantities indicated that carbon sequestration had the potential to be achieved when N = 0.5 g/kg, where active soil management measures should be taken. Furthermore, the non-equilibrium relationship between CUE and SOC was clarified through potential energy analysis, where the average deviation between predictions and actual observations of SOC is about 1.9792 g/kg. This study provides an effective framework for predicting SOC accumulation.

Statistical mechanics of phenotypic eco-evolution: From adaptive dynamics to complex diversification

The ecological and evolutionary dynamics of large populations can be addressed theoretically using concepts and methodologies from statistical mechanics. This approach has been extensively discussed in the literature, both within the realm of population genetics, which focuses on genes or “genotypes,” and in adaptive dynamics, which emphasizes traits or “phenotypes.” Following this tradition, here we construct a theoretical framework allowing us to derive “macroscopic” evolutionary equations from a general “microscopic” stochastic dynamics representing the fundamental processes of reproduction, mutation, and selection in a large community of individuals, each one characterized by its phenotypic features. Importantly, in our setup, ecological and evolutionary timescales are intertwined, which makes it particularly suitable to describe microbial communities, a timely topic of utmost relevance. The framework leads to a probabilistic description—even in the case of arbitrarily large populations—of the distribution of individuals in phenotypic space as encoded in what we call the “generalized Crow-Kimura equation” or “generalized replicator-mutator equation.” We discuss the limits in which such an equation reduces to the (deterministic) theory of “adaptive dynamics,” i.e., the standard approach to evolutionary dynamics in phenotypic space. Moreover, we emphasize the aspects of the theory that are beyond the reach of standard adaptive dynamics. In particular, by developing a simple model of a growing and competing population as an illustrative example, we demonstrate that the resulting probability distribution can undergo “dynamical phase transitions.” These transitions may involve shifts from a unimodal distribution to a bimodal distribution, generated by an evolutionary branching event, or to a multimodal distribution through a cascade of evolutionary branching events. Furthermore, our formalism allows us to rationalize these cascades using the parsimonious approach of Landau's theory of phase transitions. Finally, we extend the theory to account for finite populations and illustrate the possible consequences of the resulting stochastic or “demographic” effects. Altogether, the present framework extends and/or complements existing approaches to evolutionary and adaptive dynamics and paves the way to more systematic studies of microbial communities as well as to future developments including theoretical analyses of the evolutionary process from the general perspective of nonequilibrium statistical mechanics. Published by the American Physical Society 2024

Statistical mechanics of thermostatically controlled multizone buildings.

We study the collective phenomena and constraints associated with the aggregation of individual cooling units from a statistical mechanics perspective. These units are modeled as thermostatically controlled loads (TCLs) and represent zones in a large commercial or residential building. Their energy input is centralized and controlled by a collective unit-the air handling unit (AHU)-delivering cool air to all TCLs, thereby coupling them together. Aiming to identify representative qualitative features of the AHU-to-TCL coupling, we build a simple but realistic model and analyze it in two distinct regimes: the constant supply temperature (CST) and the constant power input (CPI) regimes. In both cases, we center our analysis on the relaxation dynamics of individual TCL temperatures to a statistical steady state. We observe that while the dynamics are relatively fast in the CST regime, resulting in all TCLs evolving around the control set point, the CPI regime reveals the emergence of a bimodal probability distribution and two, possibly strongly separated, timescales. We observe that the two modes in the CPI regime are associated with all TCLs being in the same low or high airflow states, with an occasional collective transition between the modes akin to Kramer's phenomenon in statistical physics. To the best of our knowledge, this phenomenon has been overlooked in building energy systems despite its direct operational implications. It highlights a trade-off between occupational comfort-related to zonal temperature variations-and energy consumption.

Compressed Sensing Radar Detectors Under the Row-Orthogonal Design Model: A Statistical Mechanics Perspective

Compressed Sensing Radar Detectors Under the Row-Orthogonal Design Model: A Statistical Mechanics Perspective

Additive Manufacturability Analysis of Multiscale Aperiodic Structures: A Statistical Mechanics Approach

Abstract This paper introduces heuristics based upon statistical mechanics to assist in additive manufacturability analysis of multiscale aperiodic structures. The heuristics associate structural properties at a statistical level with manufacturability. They are derived from four topological properties of the complex network representations of multiscale aperiodic structures. The validity of these heuristics is assessed in two ways. First, cross-model validation compares the manufacturability determined by these heuristics when applied to computationally designed crumpled structures and a microCT scan of the same structures when additively manufactured. Second, external validity assesses the correctness of the heuristics given design parameters that increase the potential for manufacturing errors. The results show the significance of statistical mechanics in providing insight into the additive manufacturability of multiscale aperiodic structures. The paper concludes by discussing the generality of this approach for alternative geometries and provides designers with a framework for interpreting manufacturability from a statistical mechanics perspective.

Interpreting Adversarial Examples and Robustness for Deep Learning-Based Auto-Driving Systems

Deep learning-based auto-driving systems are vulnerable to adversarial examples attacks which may result in wrong decision making and accidents. An adversarial example can fool the well trained neural networks by adding barely imperceptible perturbations to clean data. In this paper, we explore the mechanism of adversarial examples and adversarial robustness from the perspective of statistical mechanics, and propose an statistical mechanics-based interpretation model of adversarial robustness. The state transition caused by adversarial training based on the theory of fluctuation dissipation disequilibrium in statistical mechanics is formally constructed. Besides, we fully study the adversarial example attacks and training process on system robustness, including the influence of different training processes on network robustness. Our work is helpful to understand and explain the adversarial examples problems and improve the robustness of deep learning-based auto-driving systems.

Evolutionary dynamics, evolutionary forces, and robustness: A nonequilibrium statistical mechanics perspective

SignificanceEvolution through natural selection is an overwhelmingly complex process, and it is not surprising that theoretical approaches are strongly simplifying it. For instance, population genetics considers mainly dynamics of gene allele frequencies. Here, we develop a complementary approach to evolutionary dynamics based on three elements-organism reproduction, variations, and selection-that are essential for any evolutionary theory. By considering such general dynamics as a stochastic thermodynamic process, we clarify the nature and action of the evolutionary forces. We show that some of the forces cannot be described solely in terms of fitness landscapes. We also find that one force contribution can make organism reproduction insensitive (robust) to variations.

Subgrid-scale models of isotropic turbulence need not produce energy backscatter

This investigation questions the importance of inverse interscale energy fluxes, the so-called energy backscatter, for the modelling of the energy cascade in large-eddy simulations (LES) of turbulent flows. The invariance of the filtered Navier–Stokes equations to divergence-preserving transformations of the subgrid-scale (SGS) stress tensor is exploited here to explore alternative representations of the local SGS energy fluxes. Numerical optimisation procedures are applied to the SGS stress tensor – obtained by filtering isotropic turbulence flow fields – to find alternative stresses that satisfy the filtered Navier–Stokes equations, but that produce negligible backscatter. These alternative SGS stresses show that backscatter represents not a flux of energy from the subgrid to the resolved scales, but conservative spatial fluxes, and that it need not be modelled to reproduce the local energetic exchange between the resolved and the subgrid scales in LES. From the perspective of statistical mechanics, it is argued that this is a consequence of the strong statistical irreversibility of inertial-range dynamics, which precludes inverse energy cascades even in a local sense. These findings show that the energy cascade is strongly unidirectional locally, and that it can be modelled as an irreversible sink of energy, justifying the extended use of purely dissipative SGS models in LES.

Hysteresis in Combinatorial Optimization Problems

Hysteresis is a physical phenomenon reflected in macroscopic observables of materials that are subjected to external perturbations. For example, magnetic hysteresis is observed in ferromagnetic metals such as iron, nickel and cobalt in the presence of a changing external magnetic field. In this paper, we model hysteresis using combinatorial models of microscopic spin interactions, for which we invoke the top K solution framework for Ising models and their generalizations, called Weighted Constraint Satisfaction Problems (WCSPs). We show that the WCSP model with a simple "memory effect" can be used to understand hysteresis combinatorially and from the perspective of statistical mechanics. Compared to the basic Ising model, the WCSP framework allows accurate simulations of long-range and k-body interactions between the spins; and compared to other simulation frameworks, such as Monte Carlo methods, our WCSP framework has the advantage of using a principled statistical mechanics perspective. Our WCSP framework also allows us to understand hysteresis more generally in combinatorial optimization problems, with or without a connection to physically occurring phenomena.

Entropy, irreversibility and cascades in the inertial range of isotropic turbulence

This paper analyses the turbulent energy cascade from the perspective of statistical mechanics, and relates interscale energy fluxes to statistical irreversibility and information entropy production. The microscopical reversibility of the energy cascade is tested by constructing a reversible three-dimensional turbulent system using a dynamic model for the sub-grid stresses. This system, when reversed in time, develops a sustained inverse cascade towards the large scales, evidencing that the characterisation of the inertial energy cascade must consider the possibility of an inverse regime. This experiment is used to study the origin of statistical irreversibility and the prevalence of direct over inverse energy cascades in isotropic turbulence. Statistical irreversibility, a property of statistical ensembles in phase space related to entropy production, is connected to the dynamics of the energy cascade in physical space by considering the space locality of the energy fluxes and their relation to the local structure of the flow. A mechanism to explain the probabilistic prevalence of direct energy transfer is proposed based on the dynamics of the rate-of-strain tensor, which is identified as the most important source of statistical irreversibility in the energy cascade.

Non-Markovian thermal-bath-induced Brownian motion in velocity space in the presence of a magnetic field at arbitrary direction

In this work, from the perspective of statistical mechanics, the statistical properties of charged-particle motion in a microwave field and a magnetic field with a general direction described by a generalized Langevin equation subjected to an intrinsic noise with a power-law time decay correlation function have been studied. Using the general expansion theorem for the Laplace transform, the drift velocity of a charged particle in three directions can be expressed in terms of the relaxation functions. Based on the linear response theory, the expression of the complex susceptibilities, the spectral amplification, the stationary form of current density, and the power absorption have been obtained. It is noteworthy that the stochastic dynamics of a charged particle could be induced by fractional Gaussian noise. Additionally, the variances and covariances of charged particles have been studied based on the relations between relaxation functions and memory kernel functions.

Coarse-Grained Force Fields from the Perspective of Statistical Mechanics: Better Understanding of the Origins of a MARTINI Hangover

The popular MARTINI coarse-grained model is used as a test case to analyze the adherence of top-down coarse-grained molecular dynamics models (i.e., models primarily parametrized to match experimental results) to the known features of statistical mechanics for the underlying all-atom representations. Specifically, the temperature dependence of various pair distribution functions, and hence their underlying potentials of mean force via the reversible work theorem, are compared between MARTINI 2.0, Dry MARTINI, and all-atom simulations mapped onto equivalent coarse-grained sites for certain lipid bilayers. It is found that the MARTINI models do not completely capture the lipid structure seen in atomistic simulations as projected onto the coarse-grained mappings and that issues of accuracy and temperature transferability arise due to an incorrect enthalpy–entropy decomposition of these potentials of mean force. The potential of mean force for the association of two amphipathic helices in a lipid bilayer is also calculated, and especially at shorter ranges, the MARTINI and all-atom projection results differ substantially. The former is much less repulsive and hence will lead to a higher probability of MARTINI helix association in the MARTINI bilayer than occurs in the actual all-atom case. Additionally, the bilayer height fluctuation spectra are calculated for the MARTINI model, and compared to the all-atom results, it is found that the magnitude of thermally averaged amplitudes at intermediate length scales are quite different, pointing to a number of possible consequences for realistic modeling of membrane processes. Taken as a whole, the results presented here show disagreement in the enthalpic and entropic driving forces driving lateral structure in lipid bilayers as well as quantitative differences in association of embedded amphipathic helices, which can help direct future efforts to parametrize CG models with better agreement to the all-atom systems they aspire to represent.

Physics at the Molecular and Cellular Level (P@MCL): A New Curriculum for Introductory Physics

ABSTRACT In this article, I describe a new curriculum for introductory physics for the life sciences, a 2-semester sequence usually required of all biology majors. Because biology-related applications on the macroscale are complex and require mathematics beyond introductory calculus, the focus is entirely on applications from molecular and cellular biology. Topics that are more relevant for engineering have been removed, and topics relevant to biology have been added. The curriculum is designed around 2 main themes: diffusion and electric dipoles. Diffusion illustrates the concepts of conservation of momentum and energy and provides the framework for introducing entropy from the perspective of statistical mechanics. Electric dipoles illustrate the basic concepts of electromagnetic theory and provide the framework for understanding light waves and light interactions with biomolecules. These themes are supported by small computational activities to help students understand the physics without advanced mathematics. This curriculum has been piloted over the past 4 years at Michigan State University and should be applicable to many colleges and universities.

Effective Potential Description of the Interaction between Single Stem Cells and Localized Ligands

An analysis of cell-ligand interactions from a statistical mechanics perspective shows cells act as effective force-field generators, actively organizing their environment.

Possible nonequilibrium imprint in the cosmic background at low frequencies

The cosmic background radiation has been observed to deviate from the Planck law expected from a blackbody at $\sim$ 2.7 K at frequencies below $\sim 3$ GHz. We discuss the abundance of the low-energy photons from the perspective of nonequilibrium statistical mechanics. We propose a mechanism of stochastic frequency-diffusion, the counterpart to stochastic acceleration for charged particles in a turbulent plasma, to modify the standard Kompaneets equation. The resulting violation of the Einstein relation allows to take advantage of low-frequency localization and finally leads to photon cooling. The new equation predicts a frequency distribution in agreement with the absolute temperature measurements of the cosmic background radiation down to about 20 MHz, for which we offer here an updated compilation. In that sense, the so called 'space roar' we observe today is interpreted as a nonequilibrium echo of the early universe, and of nonequilibrium conditions in the primordial plasma more specifically.

Numerical Estimates of the Topological Effects in the Elasticity of Gaussian Polymer Networks and Their Exact Theoretical Description

A general elastic-thread homogenization procedure is introduced and then used to obtain direct numerical estimates of the shear modulus of Gaussian polymer networks on the basis of their representative 3D periodic microstructures. A force relaxation algorithm is employed to find equilibrium positions of the network junctions. It is shown that from a statistical mechanics perspective, this numerical algorithm can simply be regarded as a convenient method for finding the mean (averaged out over thermal fluctuations) positions of the junctions that maximize the entropy of the polymer network and therefore define the observable, equilibrium thermodynamic state of the network. As an illustration, the procedure is used to obtain estimates of the shear modulus of various model 3D periodic end-linked and vulcanized microstructures incorporating up to a million strands. The same computer microstructures are used to extract several specific topological factors operative in the considered theoretical and model descr...

Free energies of Boltzmann machines: self-averaging, annealed and replica symmetric approximations in the thermodynamic limit

Restricted Boltzmann machines (RBMs) constitute one of the main models for machine statistical inference and they are widely employed in artificial intelligence as powerful tools for (deep) learning. However, in contrast with countless remarkable practical successes, their mathematical formalization has been largely elusive: from a statistical-mechanics perspective these systems display the same (random) Gibbs measure of bi-partite spin-glasses, whose rigorous treatment is notoriously difficult.In this work, beyond providing a brief review on RBMs from both the learning and the retrieval perspectives, we aim to contribute to their analytical investigation, by considering two distinct realizations of their weights (i.e. Boolean and Gaussian) and studying the properties of their related free energies. More precisely, focusing on a RBM characterized by digital couplings, we first extend the Pastur–Shcherbina–Tirozzi method (originally developed for the Hopfield model) to prove the self-averaging property for the free energy, over its quenched expectation, in the infinite volume limit, then we explicitly calculate its simplest approximation, namely its annealed bound.Next, focusing on a RBM characterized by analogical weights, we extend Guerra’s interpolating scheme to obtain a control of the quenched free-energy under the assumption of replica symmetry (i.e. we require that the order parameters do not fluctuate in the thermodynamic limit): we get self-consistencies for the order parameters (in full agreement with the existing literature) as well as the critical line for ergodicity breaking that turns out to be the same obtained in AGS theory. As we discuss, this analogy stems from the slow-noise universality.Finally, glancing beyond replica symmetry, we analyze the fluctuations of the overlaps for a correct estimation of the (slow) noise affecting the retrieval of the signal, and by a stability analysis we recover the Aizenman–Contucci identities typical of glassy systems.

Thermodynamics and phase transition in Shapere–Wilczek fgh model: Cosmological time crystal in quadratic gravity

The Shapere–Wilczek model [1], or so called fgh model, enjoys the remarkable features of a Time Crystal (TC) that has a non-trivial time dependence in its lowest energy state (or the classical ground state). We construct a particular form of fgh model (with specified f,g,h functions) that is derived from a Mini-superspace version of a quadratic f(R,Rμν) gravity theory. Main part of the investigation deals with thermodynamic properties of such systems from classical statistical mechanics perspective. Our analysis reveals the possibility of a phase transition. Because of the higher (time) derivative nature of the model computation of the partial function is non-trivial and requires newly discovered techniques. We speculate about possible connection between our model and the Multiverse scenario.

Ensembles, dynamics, and cell types: Revisiting the statistical mechanics perspective on cellular regulation

Genetic regulatory networks control ontogeny. For fifty years Boolean networks have served as models of such systems, ranging from ensembles of random Boolean networks as models for generic properties of gene regulation to working dynamical models of a growing number of sub-networks of real cells. At the same time, their statistical mechanics has been thoroughly studied. Here we recapitulate their original motivation in the context of current theoretical and empirical research.We discuss ensembles of random Boolean networks whose dynamical attractors model cell types. A sub-ensemble is the critical ensemble. There is now strong evidence that genetic regulatory networks are dynamically critical, and that evolution is exploring the critical sub-ensemble. The generic properties of this sub-ensemble predict essential features of cell differentiation. In particular, the number of attractors in such networks scales as the DNA content raised to the 0.63 power. Data on the number of cell types as a function of the DNA content per cell shows a scaling relationship of 0.88. Thus, the theory correctly predicts a power law relationship between the number of cell types and the DNA contents per cell, and a comparable slope. We discuss these new scaling values and show prospects for new research lines for Boolean networks as a base model for systems biology.

Nonequilibrium time dynamics of genetic evolution.

Biological systems are typically highly open, nonequilibrium systems that are very challenging to understand from a statistical mechanics perspective. While statistical treatments of evolutionary biological systems have a long and rich history, examination of the time-dependent nonequilibrium dynamics has been less studied. In this paper we first derive a generalized master equation in the genotype space for diploid organisms incorporating the processes of selection, mutation, recombination, and reproduction. The master equation is defined in terms of continuous time and can handle an arbitrary number of gene loci and alleles and can be defined in terms of an absolute population or probabilities. We examine and analytically solve several prototypical cases which illustrate the interplay of the various processes and discuss the timescales of their evolution. The entropy production during the evolution towards steady state is calculated and we find that it agrees with predictions from nonequilibrium statistical mechanics where it is large when the population distribution evolves towards a more viable genotype. The stability of the nonequilibrium steady state is confirmed using the Glansdorff-Prigogine criterion.