Finite-size scaling in the canonical ensemble

We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent y1 that depends on the temperature exponent yt and the dimensionality d as y1 = −|2yt−d|. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes. 05.50.+q, 64.60.Cn, 64.60.Fr, 75.10.Hk Typeset using REVTEX 1 The central task of statistical physics is to calculate the partition function and the thermodynamic observables. Different ensembles may be employed to this purpose, which then naturally yield different forms of the thermodynamic functions. In the thermodynamic limit, the different approaches generally yield equivalent results for the relations between the thermodynamic variables. However, differences occur in finite systems. We illustrate this by means of a lattice gas, described by a finite size L, a temperature T and a third parameter governing the number of particles. In the canonical ensemble this third parameter is the particle density ρ. A calculation of an observable A in the canonical ensemble will thus, in principle, yield its expectation value 〈A〉c as a function A(T, ρ, L). In the grand ensemble, one employs the chemical potential μ as the third parameter. One may thus calculate the expectation value 〈A〉g = A(T, μ, L) as well as the grand canonical density 〈ρ〉g = ρ(T, μ, L). In the thermodynamic limit one expects in general that A(T, μ, L) = A(T, ρ(T, μ, L), L) (L → ∞), (1) for expectation values of the form 〈A〉 = ∑ Γ A(Γ)P (Γ), where P (Γ) is the probability of state Γ in the pertinent ensemble. It does not apply to quantities obtained by differentiation of observables to T , such as the specific heat. This is evident when we define A ≡ ∂A/∂T : A (g) (T, μ, L) = A (c) (T, ρ(T, μ, L), L) + ∂A ∂ρ ∂ρ ∂T . (2) The last term violates Eq. (1). It leads to the Fisher renormalization [1] effect that may affect even the exponents of leading critical singularities. In this paper, we point out that, in the canonical ensemble, new finite-size effects appear also for quantities that do not involve differentiations to T or other temperature-like variables. They appear because substituting ρ in the right-hand side of Eq. (1) is not precisely equivalent with taking the grand canonical expectation value. Namely, in the grand ensemble, ρ is still allowed to fluctuate, which is not the case in the canonical ensemble. We derive this new finite-size effect for a d-dimensional lattice gas, described by a reduced Hamiltonian H with variables σi = 0 (1) denoting the absence (presence) of a particle on lattice site i. The grand partition sum is 2 Z(T, μ, L) = 1 ∑ σ1=0 1 ∑ σ2=0 · · · 1 ∑ σN=0 exp[−H], (3) where the sum is performed independently on all N ≡ L lattice gas variables. The particle density ρ follows from differentiation of Eq. (3): ρ(T, μ, L) = 1 N ∂ lnZ ∂μ . (4) Other thermodynamic quantities A can be obtained similarly by differentiation to the conjugate parameter. This leads to the following form for the expectation value of an observable in the grand ensemble