Search

We study opinion dynamics on networks with a nontrivial community structure, assuming individuals can update their binary opinion as the result of the interactions with an external influence with strength $h\in [0,1]$ and with other individuals in the network. To model such dynamics, we consider the Ising model with an external magnetic field on a family of finite networks with a clustered structure. Assuming a unit strength for the interactions inside each community, we assume that the strength of interaction across different communities is described by a scalar $\epsilon \in [-1,1]$, which allows a weaker but possibly antagonistic effect between communities. We are interested in the stochastic evolution of this system described by a Glauber-type dynamics parameterized by the inverse temperature $\beta$. We focus on the low-temperature regime $\beta\rightarrow\infty$, in which homogeneous opinion patterns prevail and, as such, it takes the network a long time to fully change opinion. We investigate the different metastable and stable states of this opinion dynamics model and how they depend on the values of the parameters $\epsilon$ and $h$. More precisely, using tools from statistical physics, we derive rigorous estimates in probability, expectation, and law for the first hitting time between metastable (or stable) states and (other) stable states, together with tight bounds on the mixing time and spectral gap of the Markov chain describing the network dynamics. Lastly, we provide a full characterization of the critical configurations for the dynamics, i.e., those which are visited with high probability along the transitions of interest., Comment: 31 pages, 8 figures

The Liouvillian exceptional points for a quantum Markovian master equation of an oscillator in a generic environment are obtained. They occur at the points when the modified frequency of the oscillator vanishes, whereby the eigenvalues of the Liouvillian become real. In a generic system there are two parameters that modify the oscillator's natural frequency. One of the parameters can be the damping rate. The exceptional point then corresponds to critical damping of the oscillator. This situation is illustrated by the Caldeira--Leggett (CL) equation and the Markovian limit of the Hu--Paz--Zhang (HPZ) equation. The other parameter changes the oscillator's effective mass whereby the exceptional point is reached in the limit of extremely heavy oscillator. This situation is illustrated by a modified form of the Kossakowski--Lindblad (KL) equation. The eigenfunctions coalesce at the exceptional points and break into subspaces labelled by a natural number $N$. In each of the $N$-subspace, there is a $(N+1)$-fold degeneracy and the Liouvillian has a Jordan block structure of order-$(N+1)$. We obtain the explicit form of the generalized eigenvectors for a few Liouvillians. Because of the degeneracies, there is a freedom of choice in the generalized eigenfunctions. This freedom manifests itself as an invariance in the Jordan block structure under a similarity transformation whose form is obtained. We compare the relaxation of the first excited state of an oscillator in the underdamped region, critically damped region which corresponds to the exceptional point, and overdamped region using the generalized eigenvectors of the CL equation., Comment: To appear in Physica A (2023)

A transfer learning method, Domain Adversarial Neural Network (DANN), is introduced to study the phase transition of two-dimensional q-state Potts model. With the DANN, we only need to choose a few labeled configurations automatically as input data, then the critical points can be obtained after training the algorithm. By an additional iterative process, the critical points can be captured to comparable accuracy to Monte Carlo simulations as we demonstrate it for q = 3, 4, 5, 7 and 10. The type of phase transition (first or second-order) is also determined at the same time. Meanwhile, for the second-order phase transition at q=3, we can calculate the critical exponent $\nu$ by data collapse. Furthermore, compared to the traditional supervised learning, we found the DANN to be more accurate with lower cost., Comment: 30 pages, 36 figures

The two-dimensional $q$-state clock models exhibit the Berezinskii-Kosterlitz-Thouless (BKT) transition for $q\geq5$ since they are a subset of the isotropic XY model. We examine the $6$-state clock model with an anisotropic deformation. Selecting the $6$-state Potts model as a source of the deformation, the model naturally violates the discrete rotational symmetry of the clock model. We introduce the anisotropic deformation parameter $\alpha$ in the clock model interpolating the clock ($\alpha = 1$) and the Potts ($\alpha = 0$) models. We employ the corner transfer matrix renormalization group method to analyze the phase transitions on the square lattice in the thermodynamic limit. Three different phases and phase transitions are identified. The phase diagram is constructed, and we determine a tricritical point at $\alpha_{\rm c} = 0.21405(4)$ and $T_{\rm c} = 0.834017(5)$. Analyzing the latent heat and the entanglement entropy in the vicinity of the $T_{\rm c}(\alpha_{\rm c})$, we observe a single discontinuous phase transition and two BKT phase transitions meeting in the tricritical point. The tricritical point exhibits a phase transition of the second order with the critical exponents $\beta \approx 1/10$ and $\delta \approx 14$. We conjecture that an infinitesimal surrounding of the tricritical point consists of the three fundamental phase transitions, in which the first and the BKT orders gradually weaken into the second-order tricritical point.

This paper presents a toy model of a charged relativistic classical gas in flat spacetime of an arbitrary number of dimensions equipped with some ``pair production'' mechanism. Working with the microcanonical ensemble, the charge is taken to be conserved in contrast with the total number of ``particles'' plus ``antiparticles'' which varies with temperature $T$. Thermodynamics of the classical gas is detailed studied in the nonrelativistic ($Mc^{2}\gg kT$) as well as in the ultrarelativistic ($Mc^{2}\ll kT$) regimes. It is shown that, although we are dealing with classical distributions, at the ultrarelativistic regime the behavior of the gas is Planckian. We also compare the thermodynamics of the toy model with that of quantum gases., Comment: 10 pages, no figures. Replaced by the published version in "Physica A: Statistical Mechanics and its Applications"

In Stochastic Thermodynamics, heat is a random variable with a probability distribution associated. Studies of the distribution of heat are mostly in the overdamped regime and in one dimension. Here we solve the heat distribution in the underdamped regime for three different cases: the free particle, the linear potential, and the harmonic potential. All of them in arbitrary dimensions. The results are exact and generalize known results in the literature., 16 pages, 5 figures

The Blume-Emery-Griffiths model with a random crystal field is studied in a random graph architecture, in which the average connectivity is a controllable parameter. The disordered average over the graph realizations is treated by replica symmetry formalism of order parameter functions. A self-consistent equation for the distribution of local fields is derived and numerically solved by a population dynamics algorithm. The results show that the average connectivity amounts to changes in the topology of the phase diagrams. Phase diagrams for representative values of the model parameters are compared with those obtained for fully connected mean field and renormalization group approaches., Comment: 14 pages, 6 figures, accepted for publication in Physica A

The Uhlenbeck-Ford model for soft repulsion, which has only a repulsive interaction, is extended by inclusion of an attraction. This extension still allows an analytical evaluation of the virial coefficients. The integrals over the graph contributions are reduced to a combinatorial problem. We have calculated the virial coefficients to order 6 in the density. A link is made between this model and more common interactions, like the 12-6 Lennard-Jones potential.

We introduce a Gibbs Markov random field for spatial data on Cartesian grids which is based on the generalized planar rotator (GPR) model. The GPR model generalizes the recently proposed modified planar rotator (MPR) model by including in the Hamiltonian additional terms that better capture realistic features of spatial data, such as smoothness, non-Gaussianity, and geometric anisotropy. In particular, the GPR model includes up to infinite number of higher-order harmonics with exponentially vanishing interaction strength, directional dependence of the bilinear interaction term between nearest grid neighbors, longer-distance neighbor interactions, and two types of an external bias field. Hence, in contrast with the single-parameter MPR model, the GPR model features five additional parameters: the number $n$ of higher-order terms and the parameter $\alpha$ controlling their decay rate, the exchange anisotropy parameter $J^{nn}$, the further-neighbor interaction coupling $J^{fn}$, and the external field (bias) parameters $K$ (or $K'$). We present numerical tests on various synthetic data which demonstrate the effects of the respective terms on the model's prediction performance and we discuss these results in connection with the data properties., Comment: 29 pages, 9 figures

These notes are based on the lectures that I gave (virtually) at the Bruneck Summer School in 2021 on first-passage processes and some applications of the basic theory. I begin by defining what is a first-passage process and presenting the connection between the first-passage probability and the familiar occupation probability. Some basic features of first passage on the semi-infinite line and a finite interval are then discussed, such as splitting probabilities and first-passage times. I also treat the fundamental connection between first passage and electrostatics. A number of applications of first-passage processes are then presented, including the hitting probability for a sphere in greater than two dimensions, reaction rate theory and its extension to receptors on a cell surface, first-passage inside an infinite absorbing wedge in two dimensions, stochastic hunting processes in one dimension, the survival of a diffusing particle in an expanding interval, and finally the dynamics of the classic birth-death process., 30 pages, 10 figures, elsarticle format

Recent advances in operating and manipulating DNA have provided unique experimental possibilities in many fields of DNA research, especially in gene therapy. Researchers have deployed many techniques, experimental and theoretical, to study the DNA structure changes due to external perturbation. It is crucial to understand the structural and dynamical changes in the DNA molecules in a confined state to understand and control the self-assembly of DNA confined in a chamber or nano-channel for various applications. In the current manuscript, we extend the work study the effect of confinement on the thermal stability and the structural properties of duplex DNA. The present work is an extension of our previous research works. For our study we have considered a 1 BNA chain that is confined in a cylindrical geometry. How the geometry of the confinement affects the opening and other structural parameters of DNA molecule is the objective of this manuscript. We have used a statistical model(PBD model) and Molecular dynamics simulations for our purpose.

In this study, we present the original method for reconstructing the potential of interparticle interaction from statistically averaged structural data, namely, the radial distribution function of particles in many-particle system. This method belongs to a family of machine learning methods and is implemented through the differential evolution algorithm. As demonstrated for the case of the Lennard-Jones liquid taken as an example, there is no one-to-one correspondence between structure and potential of interparticle interaction of a many-particle disordered system at a certain thermodynamic state. Namely, a whole family of the Mie potentials determined by two parameters $p_{ 1 }$ and $p_{ 2 }$ related to each other according to a certain rule can reproduce properly a unique structure of the Lennard-Jones liquid at a given thermodynamic state. It is noteworthy that this family of the potentials quite correctly reproduces for the Lennard-Jones liquid the transport properties (in particular, the self-diffusion coefficient) over a temperature range as well as the dynamic structure factor, which is one of the key characteristics of the collective dynamics of particles., Comment: 29 pages, 7 figures

Resilient supply chains are often inherently dependent on the nature of their complex interconnected networks that are simultaneously multi-dimensional and multi-layered. This article presents a Supply Chain Network (SCN) model that can be used to regulate downstream relationships towards a sustainable SME using a 4-component cost function structure - Environmental (E), Demand (D), Economic (E), and Social (S). As a major generalization to the existing practice of using phenomenological interrelationships between the EDES cost kernels, we propose a complementary time varying model of a cost function, based on Lagrangian mechanics (incorporating SCN constraints through Lagrange multipliers), to analyze the time evolution of the SCN variables to interpret the competition between economic inertia and market potential. Multicriteria decision making, based on an Analytic Hierarchy Process (AHP), ranks performance quality, identifying key business decision makers. The model is first solved numerically and then validated against real data pertaining to two Small and Medium Enterprises (SMEs) from diverse domains, establishing the domain-independent nature of the model. The results quantify how increases in a production line without appropriate consideration of market volatility can lead to bankruptcy, and how high transportation cost together with increased production may lead to a break-even state. The model also predicts the time it takes a policy change to reinvigorate sales, thereby forecasting best practice operational procedure that ensures holistic sustainability on all four sustainability fronts., 8 figures, 1 table, 2 appendices. arXiv admin note: text overlap with arXiv:2003.00884

We analyse the long time tails of a charged quantum Brownian particle in a harmonic potential in the presence of a magnetic field using the Quantum Langevin Equation as a starting point. We analyse the long time tails in the position autocorrelation function, position-velocity correlation function and velocity autocorrelation function. We study these correlations for a Brownian particle coupled to the Ohmic and Drude baths, via position coordinate coupling. At finite temperatures we notice a crossover from a power-law to an exponentially decaying behaviour around the thermal time scale \frac{\hbar}{K_B T} . We analyse how the appearance of the cyclotron frequency in our study of a charged quantum Brownian particle affects the behaviour of the long time tails and contrast it with the case of a neutral quantum Brownian particle., 10 pages, 3 figures

We study the Quantum Brownian motion of a charged particle moving in a harmonic potential in the presence of an uniform external magnetic field and linearly coupled to an Ohmic bath through momentum variables. We analyse the growth of the mean square displacement of the particle in the classical high temperature domain and in the quantum low temperature domain dominated by zero point fluctuations. We also analyse the Position Response Function and the long time tails of various correlation functions. We notice some distinctive features, different from the usual case of a charged quantum Brownian particle in a magnetic field and linearly coupled to an Ohmic bath via position variables.

We develop the kinetic theory of collisionless relaxation for systems with long-range interactions in relation to the statistical theory of Lynden-Bell. We treat the multi-level case. We make the connection between the kinetic equation obtained from the quasilinear theory of the Vlasov equation and the relaxation equation obtained from a maximum entropy production principle. We propose a method to close the infinite hierarchy of kinetic equations for the phase level moments and obtain a kinetic equation for the coarse-grained distribution function in the form of a generalized Landau, Lenard-Balescu or Kramers equation associated with a generalized form of entropy [P.H. Chavanis, Physica A {\bf 332}, 89 (2004)]. This allows us to go beyond the two-level case associated with a Fermi-Dirac-type entropy. We discuss the numerous analogies with two-dimensional turbulence. We also mention possible applications of the present formalism to fermionic and bosonic dark matter halos.

In this work we propose a robust methodology to mitigate the undesirable effects caused by outliers to generate reliable physical models. In this way, we formulate the inverse problems theory in the context of Kaniadakis statistical mechanics (or $\kappa$-statistics), in which the classical approach is a particular case. In this regard, the errors are assumed to be distributed according to a finite-variance $\kappa$-generalized Gaussian distribution. Based on the probabilistic maximum-likelihood method we derive a $\kappa$-objective function associated with the finite-variance $\kappa$-Gaussian distribution. To demonstrate our proposal's outlier-resistance, we analyze the robustness properties of the $\kappa$-objective function with help of the so-called influence function. In this regard, we discuss the role of the entropic index ($\kappa$) associated with the Kaniadakis $\kappa$-entropy in the effectiveness in inferring physical parameters by using strongly noisy data. In this way, we consider a classical geophysical data-inverse problem in two realistic circumstances, in which the first one refers to study the sensibility of our proposal to uncertainties in the input parameters, and the second is devoted to the inversion of a seismic data set contaminated by outliers. The results reveal an optimum $\kappa$-value at the limit $\kappa \rightarrow 2/3$, which is related to the best results., Comment: 37 pages, 23 figures

A dilute gas of hard disks confined between two straight parallel lines is considered. The distance between the two boundaries is between one and two particle diameters, so that the system is quasi-one-dimensional. A Boltzmann-like kinetic equation, that takes into account the limitation in the possible scattering angles, is derived. It is shown that the equation verifies an -theorem implying a monotonic approach to equilibrium. The implications of this result are discussed, and the equilibrium properties are derived. Closed equations describing how the kinetic energy is transferred between the degrees of freedom parallel and perpendicular to the boundaries are derived for states that are homogeneous along the direction of the boundaries. The theoretical predictions agree with results obtained by means of Molecular Dynamics simulations. Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía US-1380729 (partially financed by FEDER 2014-2020 funds)

The classical Coulomb gas model has served as one of the most versatile frameworks in statistical physics, connecting a vast range of phenomena across many different areas. Nonequilibrium generalisations of this model have so far been studied much more scarcely. With the abundance of contemporary research into active and driven systems, one would naturally expect that such generalisations of systems with long-ranged Coulomb-like interactions will form a fertile playground for interesting developments. Here, we present two examples of novel macroscopic behaviour that arise from nonequilibrium fluctuations in long-range interacting systems, namely (1) unscreened long-ranged correlations in strong electrolytes driven by an external electric field and the associated fluctuation-induced forces in the confined Casimir geometry, and (2) out-of-equilibrium critical behaviour in self-chemotactic models that incorporate the particle polarity in the chemotactic response of the cells. Both of these systems have nonlocal Coulomb-like interactions among their constituent particles, namely, the electrostatic interactions in the case of the driven electrolyte, and the chemotactic forces mediated by fast-diffusing signals in the case of self-chemotactic systems. The results presented here hint to the rich phenomenology of nonequilibrium effects that can arise from strong fluctuations in Coulomb interacting systems, and a rich variety of potential future directions, which are discussed.

This work is dedicated to the consideration of the construction of a representation of braid group generators from vertex models with $N$-states, which provides a great way to study the knot invariant. An algebraic formula is proposed for the knot invariant when different spins $(N-1)/2$ are located on all components of the knot. The work summarizes procedure outputting braid generator representations from three-partite vertex model. This representation made it possible to study the invariant of a knot with multi-colored links, where the components of the knot have different spins. The formula for the invariant of knot with a multi-colored link is studied from the point of view of the braid generators obtained from the $R$-matrices of three-partite vertex models. The resulting knot invariant $5_2$ corresponds to the Jones polynomial and HOMFLY-PT.

We consider a gauge theory of the glass transition in the frustrated XY model being simplest model containing topologically nontrivial excitations. We describe the transition kinetics and find that the three-dimensional system exhibits the Vogel-Fulcher-Tamman criticality heralding its freezing into a spin glass. We analytically show that the system demonstrates all glass transition properties, like the logarithmic relaxation, and corresponding behavior of linear and non-linear susceptibility. The mode-coupling theory equation in the Zwanziger-Mori representation also is derived in framework of our approach. Our findings provide insights into the topological origin of glass formation, that allows to make progress in understanding glass-transition processes in more intricate systems., 18 pages. arXiv admin note: text overlap with arXiv:1712.00757

With the aid of simple analytical computations for the Ehrenfest model, we clarify some basic features of macroscopic irreversibility. The stochastic character of the model allows us to give a non-ambiguous interpretation of the general idea that irreversibility is a typical property: for the vast majority of the realizations of the stochastic process, a single trajectory of a macroscopic observable behaves irreversibly, remaining “very close” to the deterministic evolution of its ensemble average, which can be computed using probability theory. The validity of the above scenario is checked through simple numerical simulations and a rigorous proof of the typicality is provided in the thermodynamic limit.

Motivated by the study of a Mean Field Game toy model called the “seminar problem”, we consider the Fokker–Planck equation in the small noise regime for a specific drift field. This gives us the opportunity to discuss the application to diffusion problem of the WKB approach “a la Maslov (Maslov and Fedoriuk, 1981)”, making it possible to solve directly the time dependent problem in an especially transparent way.

We study the current density and transport coefficients in the fully ionized plasma with the q-distributions in nonextensive statistics and in strong magnetic field. By using the generalized Boltzmann transport equation in nonextensive statistics, we derive the current density and the expressions of the transport coefficients, including the conductivity, the thermoelectric coefficient, the Hall coefficient, and the Nernst coefficient. It is shown that these new transport coefficients has been generalized to the nonequilibrium complex plasmas with q-distributions in nonextensive statistics, which depend strongly on the q-parameters and when we take the limit q to 1, they perfectly return to those for the plasma based on a Maxwellian distribution., Comment: 10 pages,30 references. arXiv admin note: text overlap with arXiv:1807.03615

The Random Resistor cum Tunneling Network (RRTN) model was proposed from our group by considering an extra phenomenological (semi-classical) tunneling process into a classical RRN bond percolation model. We earlier reported about early-stage two inverse power-laws, followed by large time purely exponential tail in some of the RRTN macroscopic current relaxations. In this paper, we investigate on the broader perspective of current relaxation. We present here an analytical argument behind the strong convergence (irrespective of initial voltage configuration) of the bulk current towards its steady-state, mapping the problem into a special kind of Gauss-Seidel method. We find two phenomenological time-scales (referred as $\tau_t$ and $\tau_s$), those emerge from the variation of macroscopic quantities during current dynamics. We show that not both, only one of them is independent. Thus there exists a {\it single} scale in time which controls the entire dynamics., Comment: 7 Pages, 9 figures

The free rider problem is one of the most well-studied problems in economics. The solution proposed mainly is punitive in order to deter people from free riding. In this work, we introduce quantum strategies and study the problem in the thermodynamic limit by drawing analogies with the 1D Ising model. We observe that for maximum entanglement, irrespective of the payoffs, quantum strategy is the equilibrium solution, solving the free rider problem., Comment: 6 pages, 3 figures, accepted for publication in Physica A

We study the statistical properties of jump processes in a bounded domain that are driven by Poisson white noise. We derive the corresponding Kolmogorov-Feller equation and provide a general representation for its stationary solutions. Exact stationary solutions of this equation are found and analyzed in two particular cases. All our analytical findings are confirmed by numerical simulations., Comment: 16 pages, 2 figures

We analyze the properties of the majority-vote (MV) model with an additional noise in which a local spin can be changed independently of its neighborhood. In the standard MV, one of the simplest nonequilibrium systems exhibiting an order-disorder phase transition, spins are aligned with their local majority with probability $1-f$, and with complementary probability $f$, the majority rule is not followed. In the noisy MV (NMV), a random spin flip is succeeded with probability $p$ (with complementary $1-p$ the usual MV rule is accomplished). Such extra ingredient was considered by Vieira and Crokidakis [Physica A {\bf 450}, 30 (2016)] for the square lattice. Here, we generalize the NMV for arbitrary networks, including homogeneous [random regular (RR) and Erd\"os Renyi (ER)] and heterogeneous [Barabasi-Albert (BA)] structures, through mean-field calculations and numerical simulations. Results coming from both approaches are in excellent agreement with each other, revealing that the presence of additional noise does not affect the classification of phase transition, which remains continuous irrespective of the network degree and its distribution. The critical point and the threshold probability $p_t$ marking the disappearance of the ordered phase depend on the node distribution and increase with the connectivity $k$. The critical behavior, investigated numerically, exhibits a common set of critical exponents for RR and ER topologies, but different from BA and regular lattices. Finally, our results indicate that (in contrary to a previous proposition) there is no first-order transition in the NMV for large $k$., Comment: to appear in Physica A (2019)

Homogeneous melting of superheated crystals at constant energy is a dynamical process, believed to be triggered by the accumulation of thermal vacancies and their self-diffusion. From microcanonical simulations we know that if an ideal crystal is prepared at a given kinetic energy, it takes a random time t w until the melting mechanism is actually triggered. In this work we have studied in detail the statistics of t w for melting at different energies by performing a large number of Z-method simulations and applying state-of-the-art methods of Bayesian statistical inference. By focusing on a small system size and short-time tail of the distribution function, we show that t w is actually gamma-distributed rather than exponential (as asserted in a previous work), with decreasing probability near t w ∼ 0 . We also explicitly incorporate in our model the unavoidable truncation of the distribution function due to the limited total time span of a Z-method simulation. The probabilistic model presented in this work can provide some insight into the dynamical nature of the homogeneous melting process, as well as giving a well-defined practical procedure to incorporate melting times from simulation into the Z-method in order to correct the effect of short simulation times.

We develop a stochastic approach for describing the kinetics of 3D-phase transformations ruled by time-dependent correlated nucleation at solid surfaces. The kinetics is expressed as a series of correlation functions and, at odds with modeling based on Poisson statistics, it is formulated in terms of actual nucleation rate. It is shown that truncation of the series up to second order terms in correlation functions provides a very good approximation of the kinetics. The time evolution of both total amount of growing phase and surface coverage by the new phase have been determined. The theory is applied to describe progressive nucleation with parabolic growth under time dependent hard-disk correlation. This approach is suitable for describing electrochemical deposition by nucleation and growth where correlation effects are significant. In this ambit the effect of correlated nucleation on the behavior of kinetic quantities used to study electrodeposition has also been investigated.

In the current research, we investigate the concept of spontaneously nonequilibrium dimension (SND), and show that a SND-based system can break the second law of thermodynamics. The main characteristic of the SND is the inherent nonequilibrium particle crossing ratio. A locally nonchaotic energy barrier is employed to form the model system. On the one hand, when the barrier width is much smaller than the mean free path of the particles, the system cannot reach thermodynamic equilibrium; on the other hand, the nonequilibrium particle distribution allows for production of useful work in a cycle by absorbing heat from a single thermal reservoir. Such system performance is demonstrated by a Monte Carlo simulation. It should be attributed to the unbalanced cross-influence of the thermally correlated thermodynamic forces, incompatible with the conventional framework of statistical mechanics. No Maxwell's demon is involved. Similar effects may be achieved by a number of variants, e.g., when the barrier is switchable or there are distributed nonchaotic traps.

A sudden change in the macroscopic parameters of a system will cause it to depart from equilibrium. In this paper we study how a lack of change can also inform of such a departure, and allow for work extraction. Potential events that are unrealized can provide information pertaining the microstates of the system, essentially playing the role of a passive Maxwell demon, thereby allowing one to infer details about the microstate probability distribution. Here, we first qualitatively argue that this effect is present and consequential in almost every physical system, but is ignored in the standard formulation of equilibrium statistical mechanics. Then, as a case study, we quantitatively investigate the local and global thermodynamic properties of an ideal gas placed in a fragile container that nevertheless, does not burst. This non-event indicates a departure from equilibrium and allows work extraction from the system. It also leads to corrections to the heat capacity of the gas.

The inclusion of a fragmentation mechanism in population balance equations introduces complex interactions that make the analytical or even computational treatment much more difficult than for the pure aggregation case. This is specially true when variable sized fragments are allowed, because of the exponential growth in fragments size combinations with the number of monomers in the exchanges. In this contribution we present a new model that incorporates an instability threshold in the clusters, which induces arbitrary losses or gains of particles by fracture with a substantial simplification of the combinatorics of the process. The model exhibits two different regimes.

We discuss three exactly solvable spin models of geometric frustration. First, we discuss a 1-parameter subfamily of the 16 vertex model, which can be mapped to a planar Ising model and solved via Fisher-Dubed\'{a}t decorations. We then consider a 1-parameter family generalization of the Villain's fully frustrated model, which interpolates between Onsager's 2D Ising model and the Villain one. We then discuss spin ice models on a tree, which can be solved exactly using recursions \textit{a l\'{a} Bethe}.

Network growth as described by the Duplication-Divergence model proposes a simple general idea for the evolution dynamics of natural networks. In particular it is an alternative to the well known Barab\'asi-Albert model when applied to protein-protein interaction networks. In this work we derive a master equation for the node degree distribution of networks growing via Duplication and Divergence and we obtain an expression for the total number of links and for the degree distribution as a function of the number of nodes. Using algebra tools we investigate the degree distribution asymptotic behavior. Analytic results show that the network nodes average degree converges if the total mutation rate is greater than 0.5 and diverges otherwise. Treating original and duplicated node mutation rates as independent parameters has no effect on this result. However, difference in these parameters results in a slower rate of convergence and in different degree distributions. The more different these parameters are, the denser the tail of the distribution. We compare the solutions obtained with simulated networks. These results are in good agreement with the expected values from the derived expressions. The method developed is a robust tool to investigate other models for network growing dynamics., Comment: 8 Figures

All liquids are topologically disordered materials; however, the degree of disorder can vary as a result of internal fluctuations in structure and topology. These fluctuations depend on both the composition and temperature of the system. Most prior work has considered the mean values of liquid or glass properties, such as the average number of topological degrees of freedom per atom; however, the localized fluctuations in properties also play a key role in governing the macroscopic characteristics of any glass-forming system. This paper proposes a generalized approach for modeling topological fluctuations in glass-forming liquids by linking the statistical mechanics of the disordered structure to topological constraint theory. In doing so we introduce the contributions of localized fluctuations into the calculation of the topological degrees of freedoms in the network. With this approach the full distribution of properties in the disordered network can be calculated as an arbitrary function of composition, temperature, and thermal history (for the nonequilibrium glassy state). The scope of this current investigation focuses on describing topological fluctuations in liquids, concentrating on composition and temperature effects.

The purpose of the present work is to apply a recently proposed methodology to enlarge parameter domains for which optimal ratchet currents (RCs) are obtained. This task is performed by adding a suitable periodic perturbation $F_j$ on a Ratchet mapping and the procedure consists in multiplying a particular class of generic Isoperiodic Stable Structures (ISSs), since the existence of non-zero RCs is directly related to the occurrence of stable domains. By proliferating the ISSs, it is possible to: (i) postpone thermal effects that usually increase the chaotic domain and (ii) demonstrate, by using a quantitative analysis, that the area which provides optimal RCs in the two-dimensional parameter space can be enlarged about $78\%$. In addition, for some specific parameter combinations, non-zero RCs can be induced through the birth of a new attractor, which moves away as the strength of $F_j$ increases. Clearly, the methodology applied to the ratchet mapping is an efficient way to deal with unavoidable thermal effects and its consequent undesirable dynamics, specially in experimental setups. As a second general remark, we conclude that our main findings can be extended to issues related to transport problems., 8 pages, 5 figures

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., 22 pages, No figures, Standard LaTeX2e

The present work investigates the generalized extreme bounds of the coefficient of performance (COP) for the power law dissipative Carnot-like refrigerator in the presence of non-adiabatic dissipation under $\chi$ and $\dot{\Omega}$ optimization criteria. The lower and upper bounds of the COP for the low dissipation Carnot-like refrigerator under $\chi$ and $\dot{\Omega}$ optimization criteria are obtained with power law dissipation level $\delta=1$. The comparative analysis of the extreme bounds of COP at optimized $\dot{{\Omega}}$ and $\chi$ figure of merit shows the lower bound of the $\dot{\Omega}$ optimized COP is always higher than that of the $\chi$ optimized COP, while the upper bound of $\dot{\Omega}$ optimized COP is lesser than the $\chi$ optimized COP., Comment: accepted in Physica A

The existence of fluctuations of temperature has been a somewhat controversial topic in thermodynamics but nowadays it is recognized that they must be taken into account in small, finite systems. Although for nonequilibrium steady states superstatistics is becoming the \textit{de facto} framework for expressing such temperature fluctuations, some recent results put into question the idea of temperature as a phase space observable. In this work we present and explore the statistics that describes a part of an isolated system, small enough to have well-defined uncertainties in energy and temperature, but lacking a superstatistical description. These results motivate the use of the so-called fundamental temperature as an observable and may be relevant for the statistical description of small systems in physical chemistry.

We investigate a diffusion process in heterogeneous media where particles stochastically reset to their initial positions at a constant rate. The heterogeneous media is modeled using a spatial-dependent diffusion coefficient with a power-law dependence on particles' positions. We use the Green function approach to obtain exact solutions for the probability distribution of particles' positions and the mean square displacement. These results are further compared and agree with numerical simulations of a Langevin equation. We also study the first-passage time problem associated with this diffusion process and obtain an exact expression for the mean first-passage time. Our findings show that this system exhibits non-Gaussian distributions, transient anomalous diffusion (sub- or superdiffusion) and stationary states that simultaneously depend on the media heterogeneity and the resetting rate. We further demonstrate that the media heterogeneity non-trivially affect the mean first-passage time, yielding an optimal resetting rate for which this quantity displays a minimum., Comment: 8 pages, 3 figures; accepted for publication Physica A

The general formalisms of the $q$-dual statistics, the Boltzmann-Gibbs statistics, and three versions of the Tsallis statistics known as Tsallis-1, Tsallis-2, and Tsallis-3 statistics have been considered in the canonical ensemble. We have rigorously proved that the probability distribution of the Tsallis-1 statistics is invariant under the uniform energy spectrum translation at a fixed temperature. This invariance demonstrates that the formalism of the Tsallis-1 statistics is consistent with the fundamentals of the equilibrium statistical mechanics. The same results we have obtained for the probability distributions of the Tsallis-3 statistics, Boltzmann-Gibbs statistics, and $q$-dual statistics. However, we have found that the probability distribution of the Tsallis-2 statistics, the expectation values of which are not consistent with the normalization condition of probabilities, is indeed not invariant under the overall shift in energy as expected., Comment: 14 pages

We investigate the Brownian motion of a charged particle in a magnetic field. We study this in the high temperature classical and low temperature quantum domains. In both domains, we observe a transition of the mean square displacement from a monotonic behaviour to a damped oscillatory behaviour as one increases the strength of the magnetic field. When the strength of the magnetic field is negligible, the mean square displacement grows linearly with time in the classical domain and logarithmically with time in the quantum domain. We notice that these features of the mean square displacement are robust and remain essentially the same for an Ohmic dissipation model and a single relaxation time model for the memory kernel. The predictions stemming from our analysis can be tested against experiments in trapped cold ions.

Starting with the average particle distribution function for bosons and fermions for non-extensive thermodynamics , as proposed in \cite{CMP}, we obtain the corresponding density matrix operators and hamiltonians. In particular, for the bosonic case the corresponding operators satisfy a deformed bosonic algebra and the hamiltonian involves interacting terms in powers of $a^{\dagger}_ja_j$ standard creation and annihilation operators. For the unnormalized density matrix we obtain a nonlinear equation that leads to a two-parameter solution relevant to anomalous diffusion phenomena., 9 pages

In this work, we proposed a new $N$-person game in which the players can bet on two options, for example represented by two boxers. Some of the players have privileged information about the boxers and part of them can provide this information to uninformed players. However, this information may be true if the informed player is altruist or false if he is selfish. So, in this game, the players are divided in three categories: informed and altruist players, informed and selfish players, and uninformed players. By considering the matchings ($N/2$ distinct pairs of randomly chosen players) and that the payoff of the winning group follows aspects captured from two important games, the public goods game and minority game, we showed quantitatively and qualitatively how the altruism can impact on the privileged information. We localized analytically the regions of positive payoffs which were corroborated by numerical simulations performed for all values of information and altruism densities given that we know the information level of the informed players. Finally, in an evolutionary version of the game ,we showed that the gain of the informed players can get worse if we adopted the following procedure: the players increase their investment for situations of positive payoffs, and decrease their investment when negative payoffs occur., 16 pages, 6 figures

In these lecture notes, the basic principles of stochastic thermodynamics are developed starting with a closed system in contact with a heat bath. A trajectory undergoes Markovian transitions between observable meso-states that correspond to a coarse-grained description of, e.g., a biomolecule or a biochemical network. By separating the closed system into a core system and into reservoirs for ligands and reactants that bind to, and react with the core system, a description as an open system controlled by chemical potentials and possibly an external force is achieved. Entropy production and further thermodynamic quantities defined along a trajectory obey various fluctuation theorems. For describing fluctuations in a non-equilibrium steady state in the long-time limit, the concept of a rate function for large deviations from the mean behaviour is derived from the weight of a trajectory. Universal bounds on this rate function follow which prove and generalize the thermodynamic uncertainty relation that quantifies the inevitable trade-off between cost and precision of any biomolecular process. Specific illustrations are given for molecular motors, Brownian clocks and enzymatic networks that show how these tools can be used for thermodynamic inference of hidden properties of a system., Comment: Lecture notes for the international summer school "Fundamental Problems in Statistical Physics" 2017 in Bruneck

Realistic scenario can be represented by general canonical ensemble way better than the ideal one, with proper parameter sets involved. We study the Bose-Einstein condensation phenomena of liquid helium within the framework of Tsallis statistics. With a comparatively high value of the deformation parameter $q(\sim 1.4)$, the theoretically calculated value of the critical temperature($T_c$) of the phase transition of liquid helium is found to agree with the experimentally determined value ($T_c = 2.17~\rm{K}$), although they differs from each other for $q=1$ (undeformed scenario). This throws a light on the understanding of the phenomenon and connects temperature fluctuation(non-equilibrium conditions) with the interactions between atoms qualitatively. More interactions between atoms give rise to more non-equilibrium conditions which is as expected. \noindent {{\bf Keywords}: Tsallis statistics, Bose-Einstein condensation, liquid helium.}, 21 pages, 6 figures, 4 appendices

Realistic phenomena can be described more appropriately using generalized canonical ensemble, with proper parameter sets involved. We have generalized the Einstein's theory for specific heat of solid in Tsallis statistics, where the temperature fluctuation is introduced into the theory via the fluctuation parameter $q$. At low temperature the Einstein's curve of the specific heat in the nonextensive Tsallis scenario exactly lies on the experimental data points. Consequently this $q$-modified Einstein's curve is found to be overlapping with the one predicted by Debye. Considering only the temperature fluctuation effect(even without considering more than one mode of vibration is being triggered) we found that the $C_V$ vs $T$ curve is as good as obtained by considering the different modes of vibration as suggested by Debye. Generalizing the Einstein's theory in Tsallis statistics we found that a unique value of the Einstein temperature $\theta_E$ along with a temperature dependent deformation parameter $q(T)$, can well describe the phenomena of specific heat of solid i.e. the theory is equivalent to Debye's theory with a temperature dependent $\theta_D$., Comment: 20 pages, 7 figures, 3 appendices, Keywords: Tsallis statistics, temperature fluctuation, specific heat of solid, Einstein's theory, Debye's modification

The local properties of the spin one ferromagnetic Blume-Capel model defined on hierarchical lattices with dimension two and three are obtained by a numerical recursion procedure and studied as functions of the temperature and the reduced crystal-field parameter. The magnetization and the density of sites in the configuration S=0 state are carefully investigated at low temperature in the region of the phase diagram that presents the phenomenon of phase reentrance. Both order parameters undergo transitions from the ferromagnetic to the ordered paramagnetic phase with abrupt discontinuities that decrease along the phase boundary at low temperatures. The distribution of magnetization in a typical profile was determined on the transition line presenting a broad multifractal spectrum that narrows towards the fractal limit (single point) as the discontinuities of the order parameters grow towards a maximum. The amplitude of the order-parameter discontinuities and the narrowing of the multifractal spectra were used to delimit the low temperature interval for the possible locus of the tricritical point., Comment: To appear in Physica A (2017), 14 figures, 1 table, doi: 10.1016/j.physa.2017.11.156

Superstatistics is a widely employed tool of non-equilibrium statistical physics which plays an important role in analysis of hierarchical complex dynamical systems. Yet, its “canonical” formulation in terms of a single nuisance parameter is often too restrictive when applied to complex empirical data. Here we show that a multi-scale generalization of the superstatistics paradigm is more versatile, allowing to address such pertinent issues as transmutation of statistics or inter-scale stochastic behavior. To put some flesh on the bare bones, we provide a numerical evidence for a transition between two superstatistics regimes, by analyzing high-frequency (minute-tick) data for share-price returns of seven selected companies. Salient issues, such as breakdown of superstatistics in fractional diffusion processes or connection with Brownian subordination are also briefly discussed.