Microscopic entropy balance

The modification factor plays a central role in a WL simulation and has several effects. First, its presence violates microscopic detailed balance because it continuously alters the state probabilities, and hence acceptance criterion. Only for g = 0 do we obtain a true Markov sampling of our system. Furthermore, we obviously cannot resolve entropy differences which are smaller than g, yet we need the modification factor to be large enough to build up the entropy estimate in a reasonable amount of simulation time. Wang and Landau s resolution of these problems was to impose a schedule on g, in which it starts at a modest value on the order of one and decreases in stages until a value very near to zero (typically in the range 10 5-10 8). In this manner, detailed balance is satisfied asymptotically toward the end of the simulation. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

At equilibrium, the affinities vanish (A] = 0,A2 = 0). Therefore, Jrl - Jt3 = 0 and. /r2. Jr3 0 and the thermodynamic equilibrium does not require that all the reaction velocities vanish they all become equal. Under equilibrium conditions, then, the reaction system may circulate indefinitely without producing entropy and without violating any of the thermodynamic laws. However, according to the principle of detailed balance, the individual reaction velocities for every reaction should also vanish, as well as the independent flows (velocities). This concept is closely related to the principle of microscopic reversibility, which states that under equilibrium, any molecular process and the reverse of that process take place, on average, at the same rate. 

The principles of microscopic reversibility and detailed balancing imply a fundamental relationship (see Table V) between the equilibrium constant (in terms of concentrations) and the rate constants of the reversible processes. Of course, two relations are obtained, one between entropies, and one between enthalpies. 

A system at equilibrium is in a state of balance. Macroscopically, the composition of the system is unchanging whereas, at the microscopic level, change continues. The overall composition of the system does not change because, at equilibrium, a balance has been achieved between two opposing processes. The conversion of small amounts of reactants into products is always perfectly offset by the conversion of small amounts of products into reactants. This balance maintains the lowest possible value for the Gibbs energy (G) of the system and the maximum possible value for the combined entropy (S) of the system and its surroundings. Unless disturbed by an external influence, the system remains indefinitely in this equilibrium state. 

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