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45] In the periodic case (where surface effects are eliminated) one expects the leading finite size correction to come from the second term, and hence N- 1 = L-d rather than L- 1 would be the appropriate abscissa variable for linear extrapolation. 70] corrections which are exponentially small, <~T,N - ~>T,oo « exp[-L/A(T)] (1. 42) A(T) being proportional to the correlation length ~(T) of magnetization fluctuations. In that case ~>T,oo could be estimated by requesting that log I~T,N - ~T,ool vs L should give a linear variation with L.

This example illus- 21 trates a general point: one has to be very careful in the judgement if thermal equilibrium is achieved in a simulation run, since the answer may be quite different for different quantities. In the above example, S(~) for ~ large reaches equilibrium quite quickly, and so does the internal energy. Often it is said that the latter quantity is used as an indicator of thermal equilibrium of the whole system and this may be quite misleading. Next we discuss the limitation of accuracy due to finite-time averaging.

84] yields very precise estimates for systems with dimensionality close to 4, or with order parameter dimensionality close to infinity, estimates for physical systems at two and three dimensions are mostly based on extrapolations whose accuracy is not known precisely. 81] which can be appl ied directly for two- and three-dimensional lattice systems, but involve complicated numerical calculations. 85,86]. The renormalization transformation is effected by (I) grouping neighboring lattice sites into cells; (II) associating with each cell a new variable (cell spin) which reflects a collective property of the initial variables (site spins) inside the cell, (III) finding a hamiltonian for the cell spins in such a way that the partition function will be conserved.

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