Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 21
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Remark (2') has been already discussed (see point (a) above).
It is not a surprise, then, that volume entropy and metric indecomposability can
be employed to formulate a generalized Helmholtz theorem (GHT), which on one
hand extends the Helmholtz theorem to any dimension, and on the other extends the
Boltzmann-Gallavotti ideas to the continuum:
Theorem 2 (Helmholtz, generalized). Let H(p, q; V) be the Hamiltonian of a me-
chanical system with 3N degrees of freedom. Let any hyper-surfaces of constant
energy in the 6N-dim phase space F be metrically indecomposable. Let a state be
characterized by the set of quantities:
(31)
E = total energy = K + p
T = twice the time average of the kinetic energy per degree of freedom 2(K)t
3N
V = the external field
P time average of - = -a ,
Then, the differential
(32) dE + PdV
(32)
T
is exact, and the volume entropy,
(33) S(jE,V) In (E,V),
is the generating function, i.e.:
dE +PdV
(34) dSo T
T
Proof. The differential of S(E, V) is:
(35) dS= dE + dV.
Using the definition of Eq.(8):
gSc 1 8_ dz21
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Campisi, Michele. Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles, dissertation, May 2008; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc6128/m1/31/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .