Fluctuations in the critical region

Maria Adamuti‐Trache

Some critical properties for static and dynamic equilibrium superradiant models are discussed, within a Gaussian fluctuation theory. The temperature critical region and the λ-critical region are calculated and the phase diagram of the superradiant phase transition is obtained.

Yuli M. Ivanchenko, Alexander A. Lisyansky

Vitaly V. Kocharovsky, Vl. V. Kocharovsky

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.

J. P. Stark

Richard A. Ferrell

The dynamical scaling theory of phase transitions begins below the phase transition and extrapolates the frequencies, by means of the temperature continuity at a finite wave number, through the critical point into the temperature region above the transition. Applied to the ferromagnet, this approach leads to the observed five-halves critical dispersion at the Curie point. In the paramagnetic region it yields the critical slowing down of spin diffusion with a critical exponent of one-third. An alternative approach beginning in the paramagnetic region has been developed from the basic statistical mechanics of spin interaction. In the present paper, we present a simplified version of this approach, based on the fluctuation-dissipation theorem. The relevant form of this theorem is easily derived along familiar lines and involves obtaining an expression for the spin current. The spin diffusi on coefficient is then expressed in terms of the ratio of the fluctuations in spin current aud spin density. The correlation function for the current fluctuations can be evaluated by factoring (decoupled-mode approximation). The results are shown to be equivalent of those obtained by the other methods, and a simple explanation is given for the rise of the scaling function outside of the hydrodynamic region.

M. Broidioi, M Van Canneyt