For the free models of statistical mechanics on torus, exact asymptotic expansions of the free energy, the internal energy and the specific heat in the vicinity of the critical point are found. It is shown that there is direct relation between the terms of the expansion and the Kronecker's double series. The latter can be expressed in terms of the elliptic θ-functions in all orders of the asymptotic expansion. 05.50.+q, 05.70.Jk, 64.60.Cn.
The structure of avalanches in the Abelian sandpile model is analyzed. It is shown that an avalanche can be considered as a sequence of waves of dec~easing sizes. Being more elementary events, waves admit the representation in terms of the q-component Potts / model in the limit q -+ O. The decrement of waves follows the power law with the I exponent a simply related with basic exponents of the sandpile model. Using known 1\ \ exponents of the Potts model, we derive a from scaling arguments.
The Bethe ansatz method and an iterative procedure based on detailed balance are used to obtain exact results for an asymmetric avalanche process on a ring. The average velocity of particle flow, v, is derived as a function of the toppling probabilities and the density of particles, rho. As rho increases, the system shows a transition from intermittent to continuous flow, and v diverges at a critical point rho(c) with exponent alpha. The exact phase diagram of the transition is obtained and alpha is found to depend on the toppling rules.
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