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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.
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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.
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.
Bose-Einstein condensation, the macroscopic occupation of a single quantum state, appears in equilibrium quantum statistical mechanics and persists also in the hydrodynamic regime close to equilibrium. Here we show that even when a degenerate Bose gas is driven into a steady state far from equilibrium, where the notion of a single-particle ground state becomes meaningless, Bose-Einstein condensation survives in a generalized form: the unambiguous selection of an odd number of states acquiring large occupations. Within mean-field theory we derive a criterion for when a single and when multiple states are Bose selected in a non-interacting gas. We study the effect in several driven-dissipative model systems, and propose a quantum switch for heat conductivity based on shifting between one and three selected states.
We report on the realisation of a stiff magnetic trap with independently adjustable trap frequencies, $omega_z$ and $omega_r$, in the axial and radial directions respectively. This has been achieved by applying an axial modulation to a Time-averaged Orbiting Potential (TOP) trap. The frequency ratio of the trap, $omega_z / omega_r$, can be decreased continuously from the original TOP trap value of 2.83 down to 1.6. We have transferred a Bose-Einstein condensate (BEC) into this trap and obtained very good agreement between its observed anisotropic expansion and the hydrodynamic predictions. Our method can be extended to obtain a spherical trapping potential, which has a geometry of particular theoretical interest.
We present precision neutron scattering measurements of the Bose-Einstein condensate fraction, n0(T), and the atomic momentum distribution, nstar(k), of liquid 4He at pressure p =24 bar. Both the temperature dependence of n0(T) and of the width of nstar(k) are determined. The n0(T) can be represented by n0(T) = n0(0)[1-(T/T{lambda}){gamma}] with a small n0(0) = 2.80pm0.20% and large {gamma} = 13pm2 for T < T{lambda} indicating strong interaction. The onset of BEC is accompanied by a significant narrowing of the nstar(k). The narrowing accounts for 65% of the drop in kinetic energy below T{lambda} and reveals an important coupling between BEC and k > 0 states. The experimental results are well reproduced by Path Integral Monte Carlo calculations.
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