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Additional resources for Statistical Mechanics and the Foundations of Thermodynamics
Typically the following 49 s(u) = Sup S(Uo,U) is attained in an interval M = Eu~,u~, which u o can reduce to a single point, finite or infinite. e. as ^ + ~ , and the probability mass in A goes to zero very fast. all probability mass will be concentrated to M This means that in the limit. This is the basic rule in thermodynamic probability theory: The probability mass Uo(q) of a macroscopic variable the restriction that S(Uo,U) ~ ~ ~ u is maximal. in the m. can. distribution defined by will be concentrated to values of In particular, if S(Uo,U) U'o giving maximum then the law of large numbers holds: to u' o in probability.
E. e. iff. (b - a) Hence it follows by letting g'(a) (b - a) , so that g+' + ~b) # - g'(a,-b) and since gl at all both a b and not a unique at tend to a u iff. Epig g = g(a), at a iff. u g'(a) such that the directional - g(a) are different ~ gl . ) b: ~. w h i c h hence satisfy and which coincide with u+ at touches theorem then tells us that there are two supporting planes defined by c Epis and g g(a + ~b) g(a + ~b) - g(a) ~ ~g~ Hahn-Banach#s b to u = - g'(a). u is defined then then there is a direction gl ~ g'(a,b) w h e n one for all b.
V 1 8nl)dVl + (s2 ~s 2 n2 3e 2 v 2 ~--~2)dv2 = 0 ~s 2 e2 n__22 ~11) = (S 2 - 62 ~2 + 82~2 v2) " 81~I ~s 8 =~e e2 v2 8s 8~ = - ~-~ nI (S 1 - 61 ~1 + n I, n 2 ~s ~n 8W = g(8,~) = s(e,n) - 8e + 8wn , we have so the e q u i l i b r i u m condition is 81 = 62 ' gl = g2 ' or 81 = 82 , 8~Igi = B21g2 E a r l i e r we saw that b r i u m conditions 8-Ig = p = the pressure, so we arrive at the e q u i l i - 81 = 82 " Pl = P2 The i d e n t i f i c a t i o n of A2 p by a constant force H I + pAy , pA for thermal and p r e s s u r e equilibrium.