We study the factorised steady state of a general class of mass transport models in which mass, a conserved quantity, is transferred stochastically between sites. Condensation in such models is exhibited when above a critical mass density the marginal distribution for the mass at a single site develops a bump, $p_{\rm cond}(m)$, at large mass $m$. This bump corresponds to a condensate site carrying a finite fraction of the mass in the system. Here, we study the condensation transition from a different aspect, that of extreme value statistics. We consider the cumulative distribution of the largest mass in the system and compute its asymptotic behaviour. We show 3 distinct behaviours: at subcritical densities the distribution is Gumbel; at the critical density the distribution is Fréchet, and above the critical density a different distribution emerges. We relate $p_{\rm cond}(m)$ to the probability density of the largest mass in the system.
We consider the thermodynamic geometry of an ideal non-Abelian gas. We show that, for a certain value of the fractional parameter and at the relevant maximum value of fugacity, the thermodynamic curvature has a singular point. This indicates a condensation such as Bose-Einstein condensation for non-Abelian statistics and we work out the phase transition temperature in various dimensions.
We study the probability distribution $P$ of the sum of a large number of non-identically distributed random variables $n_m$. Condensation of fluctuations, the phenomenon whereby one of such variables provides a macroscopic contribution to the global probability, is discussed and interpreted in analogy to phase-transitions in Statistical Mechanics. A general expression for $P$ is derived, and its sensitivity to the details of the distribution of a single $n_m$ is worked out. These general results are verified by the analytical and numerical solution of some specific examples.
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…
Condensation is the phenomenon whereby one of a sum of random variables contributes a finite fraction to the sum. It is manifested as an aggregation phenomenon in diverse physical systems such as coalescence in granular media, jamming in traffic and gelation in networks. We show here that the same condensation scenario, which normally happens only if the underlying probability distribution has tails heavier than exponential, can occur for light-tailed distributions in the presence of additional constraints. We demonstrate this phenomenon on the sample variance, whose probability distribution c…
We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the co…
We show various aspects of finite size effects on Bose-Einstein condensation(BEC). In the first section we introduce very briefly the BEC of harmonically trapped ideal Bose gas. In the second section we theoretically argued that Bose-Einstein(B-E) statistics needs a correction for finite system at ultralow temperatures. As a corrected statistics we introduced a Tsallis type of generalized B-E statistics. The condensate fraction calculated with this generalized B-E statistics, is satisfied well with the experimental result. In the third section we show how to apply the scaling theory in an inho…
We study the dynamics of condensation for a stochastic continuous mass transport process defined on a one-dimensional lattice. Specifically we introduce three different variations of the truncated random average process. We generalize hereby the regular truncated process by introducing a new parameter $γ$ and derive a rich phase diagram in the $ρ-γ$ plane where several new phases next to the condensate or fluid phase can be observed. Lastly we use an extreme value approach in order to describe the conditions of a condensation transition in the thermodynamic limit. This leads us to a possible e…
In the thermodynamic limit the ratio of system size to thermal de Broglie wavelength tends to infinity and the volume per particle of the system is constant. Our familiar Bose-Einstein statistics is absolutely valid in the thermodynamic limit. For finite thermodynamical system this ratio as well as the number of particles is much greater than 1. However, according to the experimental setup of Bose-Einstein condensation of harmonically trapped Bose gas of alkali atoms this ratio near the condensation temperature($T_c$) typically is $\sim 32$ and at ultralow temperatures well below $T_c$ a large…
In this paper, we present an exactly solvable phase transition model in which the phase transition is purely statistically derived. The phase transition in this model is a generalized Bose-Einstein condensation. The exact expression of the thermodynamic quantity which can simultaneously describe both gas phase and condensed phase is solved with the help of the homogeneous Riemann-Hilbert problem, so one can judge whether there exists a phase transition and determine the phase transition point mathematically rigorously. A generalized statistics in which the maximum occupation numbers of differe…
By constructing the super-particle representation of the free boson gas, we propose a new statistics in which the particles are non-exclusive. This statistics can be considered as a generalization of Bose-Einstein's. The possible condensation of this statistical system is studied. It is found that the chemical potential below the condensation temperature is linearly proportional to the temperature rather than a constant. With an proper choice of the exclusion factors $γ_l$, Hadane-Wu's fractional statistics is retrieved in this representation.
The Pitman-Yor, or Chinese Restaurant Process, is a stochastic process that generates distributions following a power-law with exponents lower than two, as found in a numerous physical, biological, technological and social systems. We discuss its rich behavior with the tools and viewpoint of statistical mechanics. We show that this process invariably gives rise to a condensation, i.e. a distribution dominated by a finite number of classes. We also evaluate thoroughly the finite-size effects, finding that the lack of stationary state and self-averaging of the process creates realization-depende…
We establish an hitherto hidden connection between zero modes and instantons in the context of the Kraichanan model for passive scalar turbulent advection, that relies on the hypothesis that the production of strong gradients of the scalar is associated with Bose-like condensation of Lagrangian particles. It opens the way to the computation of scaling exponents of the N-th order structure functions of the scalar by techniques borrowed from many-body theory. To lowest order of approximation, scaling exponents are found to increase asymptotically as log N in two dimensions.
We discuss statics and dynamics of condensation in a zero-range process with compartments of limited sizes. For the symmetric dynamics the stationary state has a factorized form. For the asymmetric dynamics the steady state factorizes only for special hopping rules which allow for overjumps of fully occupied compartments. In the limit of large system size the grand canonical analysis is exact also in a condensed phase, and for a broader class of hopping rates as compared to the previously studied systems with infinite compartments. The dynamics of condensation exhibits a dynamical self-blockin…
We formulate a linear Schrodinger equation with the temperature-dependent potential for the one-particle density matrix and obtain the condensation temperature of the Bose-Einstein condensate from a bound-state condition for the Schrodinger equation both with and without the confining trap. The results are in very good agreement with those of the full statistical physics treatment. This is an alternative to the Bose-Einstein condensation in the standard ideal Bose gas treatment.
Real space condensation is known to occur in stochastic models of mass transport in the regime in which the globally conserved mass density is greater than a critical value. It has been shown within models with factorised stationary states that the condensation can be understood in terms of sums of independent and identically distributed random variables: these exhibit condensation when they are conditioned to a large deviation of their sum. It is well understood that the condensation, whereby one of the random variables contributes a finite fraction to the sum, occurs only if the underlying p…
We explore the consequences of the excluded volume interaction of hard spheres at high densities and present a theory for excited granular materials. We first demonstrate that, in the presence of gravity, the granular density crosses over from Boltzmann to Fermi statistics, when temperature is decreased in the weak excitation limit. Comparisons of numerical simulations with our predictions concerning the scaling behavior of temperature with agitation frequency, gravity and particle-diameter show satisfying agreement. Next, within the framework of the Enskog theory of hard spheres, we interpret…
In the present work we study the main physical properties of a gas of $κ$-deformed bosons described through the statistical distribution function $f_κ=Z^{-1}[\exp_κ(β({1/2}m v^2-μ))-1]^{-1}$. The deformed $κ$-exponential $\exp_κ(x)$, recently proposed in Ref. [G.Kaniadakis, Physica A {\bf 296}, 405, (2001)], reduces to the standard exponential as the deformation parameter $κ\to 0$, so that $f_0$ reproduces the Bose-Einstein distribution. The condensation temperature $T_c^κ$ of this gas decreases with increasing $κ$ value, and approaches the $^{4}He(I)-^{4}He(II)$ transition temperature $T_λ=2.…
It is shown, that Bose-Einstein statistical distributions can occur not only in quantum system, but in classical systems as well. The coherent dynamics of the system, or equivalently autocatalytic dynamics in momentum space of the system is the main reason for the Bose-Einstein condensation. A coherence is possible in both quantum and classical systems, and in both cases can lead to Bose-Einstein statistical distribution.
We extend the Bianconi-Barabási (B-B) fitness model to the non-growing complex network with fixed number of nodes and links. It is found that the statistical physics of this model makes it an appropriate representation of the Boltzmann statistics in the context of complex networks. The phase transition of this extended model is illustrated with numerical simulation and the corresponding "critical temperature" $T_c$ is identified. We note that the "non-condensation phase" in $T > T_c$ regime is different with "fit-get-rich" (FGR) phase of B-B model and that the connectivity degree distribution …
We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics, also we discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorised form. We show how this form enables one to analyse in detail condensation transitions,…
In this letter we discuss a relevance of the 3D Perfect Bose gas (PBG) condensation in extremely elongated vessels for the study of anisotropic condensate coherence and the "quasi-condensate". To this end we analyze the case of exponentially anisotropic (van den Berg) boxes, when there are two critical densities $ρ_c < ρ_m$ for a generalised Bose-Einstein Condensation (BEC). Here $ρ_c$ is the standard critical density for the PBG. We consider three examples of anisotropic geometry: slabs, squared beams and "cigars" to demonstrate that the "quasi-condensate" which exists in domain $ρ_c < ρ< ρ_m…
We consider a boson gas on the stretched horizon of the Schwartzschild and Kerr black holes. It is shown that the gas is in a Bose-Einstein condensed state with the Hawking temperature $T_c=T_H$ if the particle number of the system be equal to the number of quantum bits of space-time $ N \simeq {A}/{ł_{p}^{2}}$. Entropy of the gas is proportional to the area of the horizon $(A)$ by construction. For a more realistic model of quantum degrees of freedom on the horizon, we should presumably consider interacting bosons (gravitons). An ideal gas with intermediate statistics could be considered as a…
We find universal structure and scaling of BEC statistics and thermodynamics for mesoscopic canonical-ensemble ideal gas in a trap for any parameters, including critical region. We identify universal constraint-cut-off mechanism that makes BEC fluctuations non-Gaussian and is responsible for critical phenomena. Main result is analytical solution to problem of critical phenomena. It is derived by calculating universal distribution of noncondensate occupation (Landau function) and then universal functions for physical quantities. We find asymptotics of that solution and its approximations which …
We describe financial systems as condensates, similar to Bose-Einstein condensates, and calculate statistical distributions following from the model. The calculated distributions of investments into speculated financial assets are found equivalent to a Pareto distribution, and the calculated distributions of the price moves are found equivalent to exponentially truncated Levy distributions.
Zero-range processes with decreasing jump rates exhibit a condensation transition, where a positive fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number after adding or subtracting a subextensive excess mass of particles at the critical density. We establish a law of large numbers for the excess mass fraction in the maximum, which turns out to jump from zero to a positive value at a critical scale. Our results also include distributional limits for th…
We revisit the problem of condensation for independent, identically distributed random variables with a power-law tail, conditioned by the value of their sum. For large values of the sum, and for a large number of summands, a condensation transition occurs where the largest summand accommodates the excess difference between the value of the sum and its mean. This simple scenario of condensation underlies a number of studies in statistical physics, such as, e.g., in random allocation and urn models, random maps, zero-range processes and mass transport models. Much of the effort here is devoted …
The so-called grand canonical catastrophe of the density fluctuations in the ideal Bose gas is shown to be a particular instance of the much more general phenomenon of condensation of fluctuations, taking place in a large system, in or out of equilibrium, when a single degree of freedom makes a macroscopic contribution to the fluctuations of an extensive quantity. The pathological character of the "catastrophe" is demystified by emphasizing the connection between experimental conditions and statistical ensembles, as demonstrated by the recent realization of photon condensation under grand cano…
While equilibrium phase transitions are well described by a free-energy landscape, there are few tools to describe general features of their non-equilibrium counterparts. On the other hand, near-equilibrium free-energies are easily accessible but their full geometry is only explored in non-equilibrium, e.g. after a quench. In the particular case of a non-stationary system, however, the concepts of an order parameter and free energy become ill-defined, and a comprehensive understanding of non-stationary (transient) phase transitions is still lacking. Here, we probe transient non-equilibrium dyn…
Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 170 years ago the original motivation of thermodynamics was the description of steam engines, i.e. boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by conventional thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang-Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann-Planck stati…
Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 150 years ago the original motivation of thermodynamics was the description of steam engines, i.e. boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by canonical thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang-Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann-Planck statisti…
The thermodynamical properties of the photon-plasma system had been studied using statistical physics approach. Photons develop an effective mass in the medium thus -- as a result of the finite chemical potential -- a photon Bose-Einstein condensation can be achieved by adjusting one of the relevant parameters (temperature, photon density and plasma density) to criticality. Due to the presence of the plasma, Planck's law of blackbody radiation is also modified with the appearance of a gap below the plasma frequency where a condensation peak of coherent radiation arises for the critical system.…
