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By Jakob Yngvason, Elliott H.Lieb

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Additional info for A Guide To Entropy And The Second Law Of Thermodynamics (Princeton)

Example text

33) This important physical result was first given in [1]: the mathematical concept of the action can be identified with the thermodynamic excess work. On a fluctuational trajectory the differential excess free energy is positive and zero at a stable stationary state. We show this by considering the differential action dS dt fl = p · dx/dt|fl − H(x, p) = w(x, r)[(r · p) exp(r · p) − exp(r · p) + 1]. 34) The transition probabilities ω(x, r) are all positive and the square bracket is larger than zero except for p = 0, that is at the stable stationary state.

36) the excess differential free energy dφ is positive in general and zero at stationary states. 3 Nonlinear Multi-Variable Systems 29 Suppose we prepare this system at a given (x, y) and let it proceed along the deterministic trajectory back to the stationary state. 36) r which holds since the Hamiltonian is zero. 36) is negative unless p = 0. 37) with the equality holding only at a stationary state. Hence an excess work is required to move a system from a stable stationary state, and excess work can be done by a system relaxing towards a stable stationary state.

11) by a superscript star ‘∗ ’. The total excess work is the sum of that work in all the boxes xi Φ[{Xi }] = kT ln i xi = kt ln i Xi dXi Xi ∗ t− (Xi ) + t− Di dXi . 15) The number of X molecules in each box i is an independent variable Xi and hence the present reaction–diffusion system is isomorphic to a multivariable homogeneous system. 15) a path of integration needs to be specified because Φ is not a state function. There are two limiting cases which we can easily check. 15) reduces to x Φ(X) = kT ln X dX = kT X∗ x ln t− (X ) dX .

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A Guide To Entropy And The Second Law Of Thermodynamics (Princeton) by Jakob Yngvason, Elliott H.Lieb


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