5.2 Equilibrium

Any measurement of macroscopic property has to be made on the states at thermodynamic equilibrium of the system at a given temperature T. Equilibrium means that the average probability of finding a state n is proportional to the Boltzmann weight of that state. How to know that our system has reached that situation? One can calculate a macroscopic quantity, say magnetization M, as a function of MC time step t starting from an arbitrary configuration. As , the macroscopic quantity should reach a constant value or fluctuate slightly around a constant value. For a square lattice of size 100 x 100, magnetization, the number of up spins, are measured as a function of MC time step t and plotted in Fig.8.3. One could see that, it starts from zero as expected and reaches a steady value after 6000 time steps.

Fig.8.3: Plot of spontaneous magnetization against time t at temperature starting from a random configuration ( ).

In many cases it is possible for the system to get stuck in some metastable region of its state space for a while, giving roughly constant value of the macroscopic quantities and so appearing to have reached equilibrium. In terms of statistical Mechanics, there can be a local energy minimum in which the system can remain temporarily, and one should not mistake this as a global energy minimum, which is the region of phase space corresponds to the system in equilibrium. On may verify this by calculating a macroscopic quantity as a function of MC time steps starting from two widely different initial situation and different random number seeds. Some of the equilibrium spin configurations at different temperatures are shown below:

Fig.8.4: Equilibrium spin configuration at different temperatures around after 8000 Monte Carlo time step per spin on a square lattice of size 100 x 100.