By Jakob Yngvason, Elliott H.Lieb
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Additional info for A Guide To Entropy And The Second Law Of Thermodynamics (Princeton)
1 that the parameter α is related to the chemical potential µ through α = βµ = µ/kB T . , µ = µ T T that is, the equality of both the temperatures and the chemical potentials in both systems. Thus in equilibrium the chemical potential of hydrogen is the same, whether in gas phase or adsorbed on the catalyst, and the temperatures are the same in both phases. 4, one now considers that the system A under study is macroscopic, but much smaller that the system A (Fig. 7). Consequently, the parameters α and β of system A are almost the same as those Grand Canonical Ensemble 51 of the combined system A0 , and are very little modiﬁed when system A varies in energy or in number of particles : with respect to A the system A behaves like a heat reservoir and a particle reservoir, it dictates both its temperature β = 1/kB T and its chemical potential α = µ/kB T to A.
1 problem involves a single coordinate q1 in real space (so that the velocity q˙1 is necessarily normal to the “faces” at constant q1 ) and we determine the variation of the number of systems on the segment for the q1 coordinate [q1 , q1 + dq1 ]. ), their density is D(q1 − dq1 , . . qi , . . qÅ 3N , p1 , . . pi , . . p3Nã). Those which leave were inside the ∂ q˙1 (t) dq1 dt from the face (dark grey hatched volume, at a distance q˙1 (t) + ∂q1 area). 37) This is related to a change in the particle density in the considered volume ∂D through to dtdτ .
The statistical entropy S is then deﬁned by S = kB ln W (E) where kB = R/N is the Boltzmann constant, with R the ideal gas constant and N the Avogadro number. The general deﬁnition of the statistical entropy, valid for an ensemble average, is S = −kB pi ln pi i where pi is the probability of occurrence of the particular microstate i of the system. The expression for an isolated system is a special case of the latter deﬁnition. The statistical entropy is an extensive parameter, which is equal to zero when the microscopic state of the system is perfectly known.
A Guide To Entropy And The Second Law Of Thermodynamics (Princeton) by Jakob Yngvason, Elliott H.Lieb