By Richter M.M. (ed.), Borger E. (ed.), Oberschelp W. (ed.)
Lecture notes in arithmetic No.1104
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Extra info for Computation and Proof Theory
32) of the canonical probability measure can be obtained by maximizing the entropy under the constraint that the energy is fixed in average. Such a derivation is performed in [Balian (2007)] for instance. The constraint that the average energy of the system is fixed formalizes the idea that the system under study exchanges energy with the thermostat or energy reservoir to which it is coupled. The energy is therefore not fixed, but it has nonetheless a well-defined average value. Consider a measure which has a density ρ(q, p) with respect to the Lebesgue measure.
DqN dp1 . . dpN , so that F (z2 ) − F (z1 ) = W (z2 ) − W (z1 ). In this case, it is clear that free energy barriers are purely of energetic origin. Purely entropic barrier. Entropic barriers are often encountered in complex systems with many degrees of freedom. In this case, the system typically has enough energy to overcome the energetic barriers it can encounter, but has not, somehow, got its energy concentrated in the right modes or directions. It is expected that entropic barriers increase with the 10:37 48 World Scientific Book - 9in x 6in main˙enlib Free Energy Computations: A Mathematical Perspective y coordinate 3.
32) with respect to the Lebesgue measure dq dp. It has many similarities with the (macroscopic) thermodynamic entropy S, as shown in [Gibbs (1902)]. 49) respectively, we obtain a quantity F which should be similar to some free energy: 1 F = Eµ (H) − T Σ = − ln Zµ . 50) β Work and heat exchanges. 45) relies on a decomposition of energy exchanges into work and heat for isothermal transformations. 1. Validity and relevance of these motivations. In spite of the formal analogies highlighted above, the relationships between the microscopic definition of the free energy or entropy, and their counterparts in classical macroscopic thermodynamics are still not completely clear.
Computation and Proof Theory by Richter M.M. (ed.), Borger E. (ed.), Oberschelp W. (ed.)