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Register a new userAnalytical theory of mesoscopic Bose-Einstein condensation in ideal gas
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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 describe universal structure of critical region in terms of parabolic cylinder or confluent hypergeometric functions. Results for order parameter, statistics, and thermodynamics match known asymptotics outside critical region. We suggest 2-level and 3-level trap models and find their exact solutions in terms of cut-off negative binomial distribution (that tends to cut-off gamma distribution in continuous limit) and confluent hypergeometric distribution. We introduce a regular refinement scheme for condensate statistics approximations on the basis of infrared universality of higher-order cumulants and method of superposition and show how to model BEC statistics in actual traps. We find that 3-level trap model with matching the first 4 or 5 cumulants is enough to yield remarkably accurate results in whole critical region. We derive exact multinomial expansion for noncondensate occupation distribution and find its high temperature asymptotics (Poisson distribution). We demonstrate that critical exponents and a few known terms of Taylor expansion of universal functions, calculated previously from fitting finite-size simulations within renorm-group theory, can be obtained from presented solutions.
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Paper is published in J. Phys. A: Math. Theor. 43 (2010) 225001, doi:10.1088/1751-8113/43/22/225001. Exact analytical solution for the universal probability distribution of the order parameter fluctuations as well as for the universal statistical and thermodynamic functions of an ideal gas in the whole critical region of Bose-Einstein condensation is obtained. A universal constraint nonlinearity is found that is responsible for all nontrivial critical phenomena of the BEC phase transition. Simple analytical approximations, which describe the universal structure of the critical region in terms of confluent hypergeometric or parabolic cylinder functions, as well as asymptotics of the exact solution are derived. The results for the order parameter, all higher-order moments of BEC fluctuations, and thermodynamic quantities, including specific heat, perfectly match the known asymptotics outside critical region as well as the phenomenological renormalization-group ansatz with known critical exponents in the close vicinity of the critical point. Thus, a full analytical solution to a long-standing problem of finding a universal structure of the lambda-point for BEC in an ideal gas is found.
We present a scheme of analytical calculations determining the critical temperature and the number of condensed atoms of ideal gas Bose-Einstein condensation in external potentials with 1D, 2D or 3D periodicity. In particular we show that the width of the lowest energy band appears as the main parameter determining the critical temperature of condensation. Is obtained a very simple, proportional to 1/3 degree, regularity for this dependence. The fundamental role of tunneling in physics of condensate establishment is underscored.
We present a microscopic theory of the second order phase transition in an interacting Bose gas that allows one to describe formation of an ordered condensate phase from a disordered phase across an entire critical region continuously. We derive the exact fundamental equations for a condensate wave function and the Green functions, which are valid both inside and outside the critical region. They are reduced to the usual Gross-Pitaevskii and Beliaev-Popov equations in a low-temperature limit outside the critical region. The theory is readily extendable to other phase transitions, in particular, in the physics of condensed matter and quantum fields.
We introduce an irreversible discrete multiplicative process that undergoes Bose-Einstein condensation as a generic model of competition. New players with different abilities successively join the game and compete for limited resources. A players future gain is proportional to its ability and its current gain. The theory provides three principles for this type of competition: competitive exclusion, punctuated equilibria, and a critical condition for the distribution of the players abilities necessary for the dominance and the evolution. We apply this theory to genetics, ecology and economy.
The asymptotic (non)equivalence of canonical and microcanonical ensembles, describing systems with soft and hard constraints respectively, is a central concept in statistical physics. Traditionally, the breakdown of ensemble equivalence (EE) has been associated with nonvanishing relative canonical fluctuations of the constraints in the thermodynamic limit. Recently, it has been reformulated in terms of a nonvanishing relative entropy density between microcanonical and canonical probabilities. The earliest observations of EE violation required phase transitions or long-range interactions. More recent research on binary networks found that an extensive number of local constraints can also break EE, even in absence of phase transitions. Here we study for the first time ensemble nonequivalence in weighted networks with local constraints. Unlike their binary counterparts, these networks can undergo a form of Bose-Einstein condensation (BEC) producing a core-periphery structure where a finite fraction of the link weights concentrates in the core. This phenomenon creates a unique setting where local constraints coexist with a phase transition. We find surviving relative fluctuations only in the condensed phase, as in more traditional BEC settings. However, we also find a non-vanishing relative entropy density for all temperatures, signalling a breakdown of EE due to the presence of an extensive number of constraints, irrespective of BEC. Therefore, in presence of extensively many local constraints, vanishing relative fluctuations no longer guarantee EE.
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