Extremum Principles and the Second Law

We have already seen that all isolated systems evolve to the state of equilibrium in which the entropy reaches its maximum value. This is the basic extremum principle of thermodynamics. But we don t always deal with isolated systems. In many practical situations, the physical or chemical system under consideration is subject to constant pressure or temperature or both. In these situations the positivity of entropy change due to irreversible processes, diS 0, can also be expressed as the evolution of certain thermodynamic functions to their 

As we have seen in the previous chapter, due to irreversible processes, the entropy of an isolated system continues to increase (diS 0) until it reaches the maximum possible value. The state thus reached is the state of equilibrium. Therefore, when U and V are constant, every system evolves to a state of maximum entropy. 

The Second Law also implies that at constant S and V, every system evolves to a state of minimum energy. This can be seen as follows. We have seen that for closed systems, dU — dQ — pdV — TdgS — pdV. Because the total entropy change dS = dgS + d S, we may write dU — TdS — pdV — Td S. Since S and V are constant, dS — dV = 0. We therefore have  

in systems whose entropy is maintained at a fixed value, driven by irreversible processes, the energy evolves to the minimum possible value. 

For systems maintained at constant Tand V, a thermodynamic quantity called the Helmholtz free energy, F evolves to its minimum value. F is defined as  

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Now we introduce a different extremum principle, one that predicts the distributions of outcomes in statistical systems, such as coin flips or die rolls. This wall lead to the concept of entropy and the Second Law of thermodynamics. 

Comparative simplicity of MEIS-based computing experiments is due primarily to the simplicity of the main initial assumption of its construction on the equilibrium of all states belonging to the set of thermodynamic attainability Dt(y) and the identity of their physico-mathematical description. These states belong to the invariant manifold that contains trajectories tending to the extremum of characteristic thermodynamic function of the system and satisfying the monotonic variation of this function. The use of the mentioned assumption consistent with the second thermodynamics law allows one, as was noted, not to include in the formulation of the problem solved different more particular principles, such as the Gibbs... 

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