Classical Statistical Physics Research Articles

Effective Action in Free Probability

Recent works have explored relations between classical and quantum statistical physics on the one hand and Voiculescu's theory of free probability on the other. Motivated by these results, the present work focuses on the notion of effective action, which is closely related to the large deviation rate function in classical probability and one-particle irreducible correlation functions in quantum field theories. The central aim is to understand how it can be defined and studied in free probability. In this respect, we introduce a suitable diagrammatic formalism.

Quantum counterpart of equipartition theorem in quadratic systems

The equipartition theorem is a fundamental law of classical statistical physics, which states that every degree of freedom contributes kBT/2 to the energy, where T is the temperature and kB is the Boltzmann constant. Recent studies have revealed the existence of a quantum version of the equipartition theorem. In the present work, we focus on how to obtain the quantum counterpart of the generalized equipartition theorem for arbitrary quadratic systems in which the multimode Brownian oscillators interact with multiple reservoirs at the same temperature. An alternative method of deriving the energy of the system is also discussed and compared with the result of the quantum version of the equipartition theorem, after which we conclude that the latter is more reasonable. Our results can be viewed as an indispensable generalization of recent works on a quantum version of the equipartition theorem. Published by the American Physical Society 2024

Equivalence between finite state stochastic machine, non-dissipative and dissipative tight-binding and Schrödinger model

The mathematical equivalence between finite state stochastic machine and non-dissipative and dissipative quantum tight-binding and Schrödinger model is derived. Stochastic Finite state machine is also expressed by classical epidemic model and can reproduce the quantum entanglement emerging in the case of electrostatically coupled qubits described by von-Neumann entropy both in non-dissipative and dissipative case. The obtained results show that quantum mechanical phenomena might be simulated by classical statistical model as represented by finite state stochastic machine. It includes the quantum like entanglement and superposition of states. Therefore coupled epidemic models expressed by classical systems in terms of classical physics can be the base for possible incorporation of quantum technologies and in particular for quantum like computation and quantum like communication. The classical density matrix is derived and described by the equation of motion in terms of anticommutator. Existence of Rabi like oscillations is pointed in classical epidemic model. Furthermore, the existence of Aharonov–Bohm effect in quantum systems (Becker et al., 2019) can also be reproduced by the classical epidemic model or in broader sense by finite state stochastic machine. Every quantum system made from quantum dots and described by simplistic tight-binding model by use of position-based qubits can be effectively described by classical statistical model encoded in finite stochastic state machine with very specific structure of S matrix that has twice bigger size as it is the case of quantum matrix Hamiltonian. Furthermore, the description of linear and non-linear stochastic finite state machine is mapped to tight-binding and Schrödinger model. The concept of N dimensional complex time is incorporated into tight-binding model, so the description of dissipation in most general case is possible. Obtained results help in approximation of non-dissipative or dissipative quantum mechanical phenomena by classical statistical physics expressed by finite state stochastic machine. Furthermore, the equivalence of Wannier tight-binding dissipative or non-dissipative formalism to dissipative or non-dissipative Schrödinger formalism and to finite state stochastic machine was shown.

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.

Synthesis and physicochemical characterization of new calcined layered double hydroxide Mg Zn Co Al-CO3; classical modeling and statistical physics of nitrate adsorption

This research focuses on the study of a new material, a layered double hydroxide, which was synthesized using a co-precipitation method at pH constant and, then, tested for its retention of nitrate anions. The LDH was prepared from four metal cations Mg-Zn-Co-Al to produce the quadratic material mentioned, Mg Zn Co Al-CO3, which was calcined to obtain metal oxides. Different techniques of characterization were considered, namely FTIR, X-ray diffraction, BET, SEM and EDX. Adsorption experiments for nitrate were carried out to test the effect of contact time, adsorbent mass, pH values, mass effect, and initial surfactant concentrations. When equilibrium was reached after 300 min (R2 = 0.998), the Pseudo-second order model was found to fit accurately the kinetic data. The intraparticle diffusion model recorded a rate constant of between 0.277 and 0.757 (g/mg. min−0.5) and the Sips adsorption isotherm best fitted the experimental data obtained. Maximum adsorption capacity for nitrate was found to be 55.97 mg g−1 which was in accordance with the experimental results. Thermodynamic parameters indicated that the adsorption process was spontaneous and endothermic. The values of ΔH˚ and ΔS˚ of the adsorption process were 14.81 and 59.04 kJ mol−1, respectively. The low value of ΔH˚ (<40 kJ mol−1) indicated that the adsorption process was associated with physical forces. Finally, for a deeper insight into the nitrate adsorption mechanism, the statistical physics model was considered to quantify the number of adsorbed nitrate molecules per site, the anchorage number, the receptor sites density, the adsorbed quantity at saturation, the concentration at half saturation and the molar adsorption energy. A statistical physics model, with two binding sites, proved to be the best option to characterize and interpret the removal of nitrate on the tested LDH.

Fe3O4 Nanoparticles Loaded Bentonite/Sawdust Interface for the Removal of Methylene Blue: Insights into Adsorption Performance and Mechanism via Experiments and Theoretical Calculations

Herein, magnetite nanoparticles (MNPs) synthesized from altered basalt were used to develop a composite with H2O2–activated bentonite (BE) and fibrous sawdust (SD). The as-prepared BE/SD–MNPs were characterized by FTIR, FESEM, TEM, TGA, DSC, and Zeta potential techniques and utilized as an effective multifunctional composite for removing methylene blue (MB). The adsorption isotherms of MB at 25–55 °C were analyzed via kinetics, classical, and advanced statistical physics models. Theoretically, the pseudo-second-order of kinetics and the Freundlich isotherms model fit the experimental data well without microscopically clarifying the adsorption mechanism. Studying a multilayer model’s steric and energetic parameters was a reliable approach to understanding the MB uptake mechanism at the molecular scale. Sterically, the removed MB molecules offered a combination of horizontal and vertical geometry (i.e., mixed orientation). The MB adsorption capacity at saturation (Qsat) increased from 829 to 849 mg/g with temperature, suggesting endothermic interactions. Energetically, the MB uptake by BE/SD–MNPs was controlled by physical interactions (i.e., adsorption energy &lt; 20 kJ/mol). The BE/SD–MNPs retained more than 85% of the MB uptake after five adsorption-desorption rounds. Overall, this study aimed to understand the MB adsorption mechanism using a magnetic clays/lignocellulosic interface such as the utilized BE/SD–MNPs composite as a promising strategy in wastewater remediation.

Insights Into the Mn(VII) and Cr(VI) Adsorption Mechanisms on Purified Diatomite/MCM-41 Composite: Experimental Study and Statistical Physics Analysis.

In this study, a purified diatomite (PD) with a concentration of diatom frustules more than 92% SiO2 was utilized to synthesize a composite of MCM-41 silica under hydrothermal conditions. The as-synthesized PD/MCM-41 composite was characterized and tested as an adsorbent for the removal of Cr(VI) and Mn(VII) ions from aqueous solution. Results of X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), and Fourier-transform infrared spectroscopy (FTIR) revealed that the diatom frustules of the PD were coated with MCM-41 mesoporous silica. Experimental isotherms of Cr(VI) and Mn(VII) adsorption were fitted to classical and advanced statistical physics models at 25°C–55°C and pH 3. The Langmuir model estimated monolayer adsorption capacities ranging from 144.1 to 162.2 mg/g for Cr(VI) and 166.2 to 177.0 mg/g for Mn(VII), which improved with increasing the solution temperature. Steric and energetic parameters obtained from a monolayer adsorption model with one adsorption site was utilized to explain the adsorption mechanism at a microscopic level. The number of Cr(VI) and Mn(VII) ions adsorbed on PD/MCM-41 active site (n) were 1.25–1.27 for Cr(VI) and 1.27–1.32 for Mn(VII), thus suggesting multi-interaction mechanisms. The density of PD/MCM-41 active sites (D M) was a key parameter to explain the adsorption of these heavy metals. The adsorbed quantities were maximum at 55°C, thus obtaining 102.8 and 110.7 mg/g for Cr(VI) and Mn(VII), respectively. Cr(VI) and Mn(VII) adsorption energies ranged from 18.48 to 26.70 kJ/mol and corresponded to an endothermic adsorption with physical forces. Entropy, free enthalpy, and internal energy associated to the adsorption of Cr(VI) and Mn(VII) ions were calculated, thus indicating that the removal of these pollutants was spontaneous. Overall, this article offers new interpretations for the Cr(VI) and Mn(VII) adsorption mechanisms on PD/MCM-41 composite, which are relevant to contribute to the development of effective water treatment processes.

The quantum taste of hydrogen

Electronic properties of materials are dominated by quantum effects, but nuclei, being much heavier, are usually treated as classical particles. This approximation, although tremendously convenient, is not always valid, even in close to ambient pressure and temperature conditions, especially when light nuclei such as hydrogen are involved. Zero point energy and proton tunneling can be relevant. Isotopic effects, obtained by replacing hydrogen with deuterium, are observed experimentally and are a clear indication of Nuclear Quantum Effects (NQE) since mean values obtained through classical statistical physics do not depend on mass. Introducing NQEs into simulations at an acceptable computational cost raises fundamental questions and yields subtle and unexpected results.

Disorder-induced quantum-to-classical transition, or how the world becomes classical

Decoherence theory explains how quantum mechanics gives rise to classical mechanics through the entanglement of a quantum system’s evolution with the degrees of freedom of the environment. The present article explores another pathway from the quantum to the classical behaviour. We consider a spinless particle interacting with a disordered landscape of potential energy. The matterwave evolution is handled within time-dependent quantum statistical mechanics, in which the wave function is replaced by a Wigner function defined in position-momentum space. Upon zooming out to scales exceeding the correlation length of the disorder, it is found that the description only involves the state populations as defined in classical statistical physics. Quantum coherence effects are significant only over smaller spatial scales, where they give rise to a noise superimposing on the classical description. The waning of coherence, which reflects the emergence of classicality, is due to the multiple scattering of matter waves; and the framework may be viewed as a stochastic wave mechanics.

Coarse-graining and symplectic non-squeezing

We address aspects of coarse-graining in classical Statistical Physics from the viewpoint of the symplectic non-squeezing theorem. We make some comments regarding the implications of the symplectic non-squeezing theorem for the BBGKY hierarchy. We also see the cubic cells appearing in coarse-graining as a direct consequence of the uniqueness of Hofer’s metric on the group of Hamiltonian diffeomorphisms of the phase space.

Comment on “Review of experimental studies of secondary ice production” by Korolev and Leisner (2020)

Abstract. This is a comment on the review by Korolev and Leisner (2020, hereafter KL2020). The only two laboratory/field studies ever to measure the breakup in ice–ice collisions for in-cloud conditions were negatively criticised by KL2020, as were our subsequent theoretical and modelling studies informed by both studies. First, hypothetically, even without any further laboratory experiments, such theoretical and modelling studies would continue to be possible, based on classical mechanics and statistical physics. They are not sensitive to the accuracy of lab data for typical situations, partly because the nonlinear explosive growth of ice concentrations continues until some maximum concentration is reached. To a degree, the same final concentration is expected regardless of the fragment number per collision. Second, there is no evidence that both lab/field observational studies characterising fragmentation in ice–ice collisions are either mutually conflicting or erroneous such that they cannot be used to represent this breakup in numerical models, contrary to the review. The fact that the ice spheres of one experiment were hail sized (2 cm) is not a problem if a universal theoretical formulation, such as ours, with fundamental dependencies, is informed by it. Although both lab/field studies involved head-on collisions, rotational kinetic energy for all collisions generally is only a small fraction of the initial collision kinetic energy (CKE) anyway. Although both lab/field experiments involved fixed targets, that is not a problem since the fixing of the target is represented via CKE in any energy-based formulation such as ours. Finally, scaling analysis suggests that the breakup of ice during sublimation can make a significant contribution to ice enhancement in clouds, again contrary to the impression given by the review.

Perspective—On the thermodynamics of perfect unconditional security

A secure key distribution (exchange) scheme is unconditionally secure if it is unbreakable against arbitrary technological improvements of computing power and/or any development of new algorithms. There are only two families of experimentally realized and tested unconditionally secure key distribution technologies: quantum key distribution (QKD), the base of quantum cryptography, which utilizes quantum physical photonic features, and the Kirchhoff-Law–Johnson-Noise (KLJN) system that is based on classical statistical physics (fluctuation–dissipation theorem). The focus topic of this paper is the thermodynamical situation of the KLJN system. In all the original works, the proposed KLJN schemes required thermal equilibrium between the devices of the communicating parties to achieve perfect security. However, Vadai et al., in (Nature) Sci. Rep. 5, 13653 (2015) show a modified scheme, where there is a non-zero thermal noise energy flow between the parties, yet the system seems to resist all the known attack types. We introduce an attack type against their system. The attack utilizes coincidence events between the line current and voltages. We show that there is a non-zero information leak toward the Eavesdropper, even under idealized conditions. As soon as the thermal equilibrium is restored, the system becomes perfectly secure again. In conclusion, perfect unconditional security requires thermal equilibrium.

Nonlinear fluctuation behaviors of complex voter financial price dynamics on small-world network

To simulate the price fluctuation dynamics of financial markets, a novel financial price model is developed by the voter dynamic system on the Watts-Strogtz small-world network and the random jump process. The voter system is a classical statistical physics system, which describes the dynamics of voters’ attitudes towards a certain topic in the mutual influence. The Watts-Strogtz small-world network is a special kind of complex networks, which can be used to study the transmission dynamics of different things in complex and real-world systems. The paper first attempts to use the voter dynamic system on the small-world network to reproduce the micro-mechanism of price fluctuations caused by the interaction among different investors in financial markets, where investors can potentially disseminate information and interact via additional long-distance contacts. Moreover, considering that external macro environments have the impact on price fluctuations in financial markets, this paper introduces the random jump process in the price model. The effectiveness of the proposed model can be verified by comparing price returns generated by the model with returns of several important stock indexes in terms of nonlinear fluctuation behaviors. First, some statistical behaviors of the fluctuation dynamics are explored, including distribution characteristics and autocorrelation. Moreover, based on the ensemble empirical mode decomposition method, multifractal behaviors and complexity behaviors of returns and the first three intrinsic mode functions are investigated. The empirical results show that the dynamical model can well simulate these nonlinear fluctuation behaviors of real markets.

Extended harmonic mapping connects the equations in classical, statistical, fluid, quantum physics and general relativity

One potential pathway to find an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that of classical mechanics, fluid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections reflect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold.

Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories.

We investigate theoretically the emergence of classical statistical physics in a finite quantum system that is either totally isolated or otherwise subjected to a quantum measurement process. We show via a random matrix theory approach to nonintegrable quantum systems that the set of outcomes of the measurement of a macroscopic observable evolve in time like stochastic variables, whose variance satisfies the celebrated Einstein relation for Brownian diffusion. Our results show how to extend the framework of eigenstate thermalization to the prediction of properties of quantum measurements on an otherwise closed quantum system. We show numerically the validity of the random matrix approach in quantum chain models.

Invariance Principle for the Random Lorentz Gas\u2014Beyond the Boltzmann-Grad Limit

We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density varrho , in the limit varrho rightarrow infty , rrightarrow 0, varrho r^{2}rightarrow 1 up to time scales of order T=o(r^{-2}left| {log r}right| ^{-2}). To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (Phys Rev 185:308–322, 1969, Nota Interna Univ di Roma 358, 1970, Statistical mechanics. A short treatise. Theoretical and mathematical physics series, Springer, Berlin, 1999), Spohn (Commun Math Phys 60:277–290, 1978, Rev Mod Phys 52:569–611, 1980) and Boldrighini–Bunimovich–Sinai (J Stat Phys 32:477–501, 1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski–Ryzhik (Commun Math Phys 263:277–323, 2006), respectively, Erdős–Salmhofer–Yau (Acta Math 200:211–277, 2008, Commun Math Phys 271:1–53, 2007). However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2) The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation—expressed in terms of the kinetic time scale—is much longer and fully explicit.

Analytical percolation theory for topological color codes under qubit loss

Quantum information theory has shown strong connections with classical statistical physics. For example, quantum error correcting codes like the surface and the color code present a tolerance to qubit loss that is related to the classical percolation threshold of the lattices where the codes are defined. Here we explore such connection to study analytically the tolerance of the color code when the protocol introduced in [Phys. Rev. Lett. $\textbf{121}$, 060501 (2018)] to correct qubit losses is applied. This protocol is based on the removal of the lost qubit from the code, a neighboring qubit, and the lattice edges where these two qubits reside. We first obtain analytically the average fraction of edges $ r(p) $ that the protocol erases from the lattice to correct a fraction $ p $ of qubit losses. Then, the threshold $ p_c $ below which the logical information is protected corresponds to the value of $ p $ at which $ r(p) $ equals the bond-percolation threshold of the lattice. Moreover, we prove that the logical information is protected if and only if the set of lost qubits does not include the entire support of any logical operator. The results presented here open a route to an analytical understanding of the effects of qubit losses in topological quantum error codes.

On entropy for general quantum systems

In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory.

Ott-Antonsen ansatz truncation of a circular cumulant series

The cumulant representation is common in classical statistical physics for variables on the real line and the issue of closures of cumulant expansions is well elaborated. The case of phase variables significantly differs from the case of linear ones; the relevant order parameters are the Kuramoto-Daido ones but not the conventional moments. One can formally introduce `circular' cumulants for Kuramoto-Daido order parameters, similar to the conventional cumulants for moments. The circular cumulant expansions allow to advance beyond the Ott-Antonsen theory and consider populations of real oscillators. First, we show that truncation of circular cumulant expansions, except for the Ott-Antonsen case, is forbidden. Second, we compare this situation to the case of the Gaussian distribution of a linear variable, where the second cumulant is nonzero and all the higher cumulants are zero, and elucidate why keeping up to the second cumulant is admissible for a linear variable, but forbidden for circular cumulants. Third, we discuss the implication of this truncation issue to populations of quadratic integrate-and-fire neurons [E. Montbri\'o, D. Paz\'o, A. Roxin, Phys. Rev. X, vol. 5, 021028 (2015)], where within the framework of macroscopic description, the firing rate diverges for any finite truncation of the cumulant series, and discuss how one should handle these situations. Fourth, we consider the cumulant-based low-dimensional reductions for macroscopic population dynamics in the context of this truncation issue. These reductions are applicable, where the cumulant series exponentially decay with the cumulant order, i.e., they form a geometric progression hierarchy. Fifth, we demonstrate the formation of this hierarchy for generic distributions on the circle and experimental data for coupled biological and electrochemical oscillators.

Generalization from correlated sets of patterns in the perceptron

Generalization is a central aspect of learning theory. Here, we propose a framework that explores an auxiliary task-dependent notion of generalization, and attempts to quantitatively answer the following question: given two sets of patterns with a given degree of dissimilarity, how easily will a network be able to ‘unify’ their interpretation? This is quantified by the volume of the configurations of synaptic weights that classify the two sets in a similar manner. To show the applicability of our idea in a concrete setting, we compute this quantity for the perceptron, a simple binary classifier, using the classical statistical physics approach in the replica-symmetric ansatz. In this case, we show how an analytical expression measures the ‘distance-based capacity’, the maximum load of patterns sustainable by the network, at fixed dissimilarity between patterns and fixed allowed number of errors. This curve indicates that generalization is possible at any distance, but with decreasing capacity. We propose that a distance-based definition of generalization may be useful in numerical experiments with real-world neural networks, and to explore computationally sub-dominant sets of synaptic solutions.