Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2511.06287

Ordering in statistical systems on the way to the thermodynamic limit

Vyacheslav I. Yukalov, Elizaveta P. Yukalova

Published Nov 9, 2025• 0 citations
arXivHTMLPDF

Abstract

It is well known that the mathematically accurate description of ordering and related symmetry breaking in statistical systems requires to consider the thermodynamic limit. But the order does not appear from nowhere, and yet before the thermodynamic limit is reached, there should exist some kind of preordering that appears and grows in the process of increasing the system size. The quantitative description of growing order, under the growing system size, is developed by introducing the notion of {\it order indices}. The rigorous proof of the phase transition existence is a separate difficult problem that is not the topic of the present paper. We illustrate the approach resorting to several models in the mean-field approximation, which makes it possible to demonstrate the notion of order indices for finite systems in a clear way. We show how the order grows on the way to the thermodynamic limit for Bose-Einstein condensation, arising superconductivity, magnetization, and crystallization phenomena.

DOI
10.48550/arxiv.2511.06287

Related papers

Matrix Order Indices in Statistical Mechanics

Vyacheslav I. Yukalov

A new notion is introduced of matrix order indices which relate the matrix norm and its trace. These indices can be defined for any given matrix. They are especially important for matrices describing many-body systems, equilibrium as well as nonequilibrium, for which the indices present a quantitative measure of the level of ordering. They characterize not only the long-range order, but also midrange order. In the latter case, when order parameters do not exist, the matrix indices are well defined, providing an explicit classification of various mid-range orders. The matrix order indices are suitable for describing phase transitions with both off-diagonal and diagonal order. Contrary to order parameters whose correct definition requires the thermodynamic limit, the matrix indices do not necessarily need the latter. Because of this, such indices can distinguish between different phases of finite systems, thus, allowing for the classification of crossover phase transitions. PACS: 02.10.Yn; 05.70.Fh; 64.60.Cn

•Jan 1, 2002•15 citations

Complexity of finite statistical systems

Vyacheslav I. Yukalov, Elizaveta P. Yukalova

Abstract The paper advances a novel notion–Complexity of Statistical Systems which is applicable to both, infinite systems in the thermodynamic limit as well as to finite systems, such as the systems of trapped atoms and molecules. The notion takes into account the level of ordering and the strength of correlations in the system. Illustrations are given by presenting calculations of complexity for finite systems with Bose–Einstein condensate and for superfluid systems.

Laser Physics Letters•Oct 1, 2026•0 citations

Phase transitions in finite systems

Raj K. Pathria

Mathematical singularities, which are known to be at the heart of phase transitions and are responsible for making the thermodynamic functions of the given system nonanalytic, are a consequence of the thermodynamic limit, viz. N and V → ∞ with N/V staying constant. When a system containing a finite number of particles and confined to a restricted geometry undergoes a phase transition, these singularities get rounded off, with the result that all thermodynamic functions become analytic and vary smoothly with the relevant parameters of the problem. Theoretical analysis of such situations requires the use of special mathematical techniques which may vary drastically from case to case.In the present communication we report the results of a rigorous analysis of the problem of "Bose–Einstein condensation in restricted geometries", which has been carried out by making an extensive use of the Poisson summation formula. Particular emphasis is laid on the growth of the condensate fraction [Formula: see text] as the temperature of the system is lowered, and on the influence of the boundary conditions imposed on the wave functions of the particles. The relevance of these results, in relation to the scaling theory of finite size effects, is also discussed.

Canadian Journal of Physics•Feb 1, 1983•38 citations

Introduction

Igor F. Herbut

Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.

Cambridge University Press eBooks•Jan 4, 2007•0 citations

Approaching Bose–Einstein condensation

L. Ferrari

Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.

European Journal of Physics•Oct 4, 2011•5 citations

Classification theory for anequilibrium phase transitions

Rudolf Hilfer A

The paper introduces a classification of phase transitions in which each transition is characterized through its generalized order and a slowly varying function. This characterization is shown to be applicable in statistical mechanics as well as in thermodynamics albeit for different mathematical reasons. By introducing the block ensemble limit the statistical classification is based on the theory of stable laws from probability theory. The block ensemble limit combines scaling limit and thermodynamic limit. The thermodynamic classification on the other hand is based on generalizing Ehrenfest's traditional classification scheme. Both schemes imply the validity of scaling at phase transitions without the need to invoke renormalizaton-group arguments. The statistical classification scheme allows derivation of a form of finite-size scaling for the distributions of statistical averages while the thermodynamic classification gives rise to multiscaling of thermodynamic potentials. The different nature of the two classification theories is also apparent from the fact that the generalized thermodynamic order is unbounded while the statistical order is restricted to values less than 2. This fact is found to be related to the breakdown of hyperscaling relations. Both classification theories predict the possible existence of phase transitions having orders less than unity. Such transitions are termed anequilibrium transitions.Systems near anequilibrium transitions cannot be described by conventional equilibrium thermodynamics or equilibrium statistical mechanics because of very strong fluctuations. Anequilibrium transitions are found to exist in statistical-mechanical model systems. The identification of the Lagrange parameter \ensuremath{\beta} in the canonical ensemble becomes invalid if a reservoir and a system of the same substance are in thermal contact and anequilibrium transitions are present. Based on the ergodic hypothesis and the theory of convolution semigroups it is shown that near anequilibrium transitions the equations of motion for macroscopic observables of infinite systems may involve modified time derivatives as generators of the macroscopic time evolution. The general solution to the modified equations of motion exhibits very slow dynamics as frequently observed in a nonequilibrium experiment.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1993•59 citations
It is well known that the mathematically accurate description of ordering and related symmetry breaking in statistical systems requires to consider the thermodynamic limit. But the order does not appear from nowhere, and yet before the thermodynamic limit is reached, there should exist some kind of preordering that appears and grows in the process of increasing the system size. The quantitative description of growing order, under the growing system size, is developed by introducing the notion of {\it order indices}. The rigorous proof of the phase transition existence is a separate difficult problem that is not the topic of the present paper. We illustrate the approach resorting to several models in the mean-field approximation, which makes it possible to demonstrate the notion of order indices for finite systems in a clear way. We show how the order grows on the way to the thermodynamic limit for Bose-Einstein condensation, arising superconductivity, magnetization, and crystallization phenomena.
A new notion is introduced of matrix order indices which relate the matrix norm and its trace. These indices can be defined for any given matrix. They are especially important for matrices describing many-body systems, equilibrium as well as nonequilibrium, for which the indices present a quantitative measure of the level of ordering. They characterize not only the long-range order, but also midrange order. In the latter case, when order parameters do not exist, the matrix indices are well defined, providing an explicit classification of various mid-range orders. The matrix order indices are suitable for describing phase transitions with both off-diagonal and diagonal order. Contrary to order parameters whose correct definition requires the thermodynamic limit, the matrix indices do not necessarily need the latter. Because of this, such indices can distinguish between different phases of finite systems, thus, allowing for the classification of crossover phase transitions. PACS: 02.10.Yn; 05.70.Fh; 64.60.Cn
Mathematical singularities, which are known to be at the heart of phase transitions and are responsible for making the thermodynamic functions of the given system nonanalytic, are a consequence of the thermodynamic limit, viz. N and V → ∞ with N/V staying constant. When a system containing a finite number of particles and confined to a restricted geometry undergoes a phase transition, these singularities get rounded off, with the result that all thermodynamic functions become analytic and vary smoothly with the relevant parameters of the problem. Theoretical analysis of such situations requires the use of special mathematical techniques which may vary drastically from case to case.In the present communication we report the results of a rigorous analysis of the problem of "Bose–Einstein condensation in restricted geometries", which has been carried out by making an extensive use of the Poisson summation formula. Particular emphasis is laid on the growth of the condensate fraction [Formula: see text] as the temperature of the system is lowered, and on the influence of the boundary conditions imposed on the wave functions of the particles. The relevance of these results, in relation to the scaling theory of finite size effects, is also discussed.
Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.
The paper introduces a classification of phase transitions in which each transition is characterized through its generalized order and a slowly varying function. This characterization is shown to be applicable in statistical mechanics as well as in thermodynamics albeit for different mathematical reasons. By introducing the block ensemble limit the statistical classification is based on the theory of stable laws from probability theory. The block ensemble limit combines scaling limit and thermodynamic limit. The thermodynamic classification on the other hand is based on generalizing Ehrenfest's traditional classification scheme. Both schemes imply the validity of scaling at phase transitions without the need to invoke renormalizaton-group arguments. The statistical classification scheme allows derivation of a form of finite-size scaling for the distributions of statistical averages while the thermodynamic classification gives rise to multiscaling of thermodynamic potentials. The different nature of the two classification theories is also apparent from the fact that the generalized thermodynamic order is unbounded while the statistical order is restricted to values less than 2. This fact is found to be related to the breakdown of hyperscaling relations. Both classification theories predict the possible existence of phase transitions having orders less than unity. Such transitions are termed anequilibrium transitions.Systems near anequilibrium transitions cannot be described by conventional equilibrium thermodynamics or equilibrium statistical mechanics because of very strong fluctuations. Anequilibrium transitions are found to exist in statistical-mechanical model systems. The identification of the Lagrange parameter \ensuremath{\beta} in the canonical ensemble becomes invalid if a reservoir and a system of the same substance are in thermal contact and anequilibrium transitions are present. Based on the ergodic hypothesis and the theory of convolution semigroups it is shown that near anequilibrium transitions the equations of motion for macroscopic observables of infinite systems may involve modified time derivatives as generators of the macroscopic time evolution. The general solution to the modified equations of motion exhibits very slow dynamics as frequently observed in a nonequilibrium experiment.