By Harrison D.M.
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Extra info for About Mass-Energy Equivalence
20) and one recognizes f3 = 1/ kT and the partition function (sum over states) Z := e-/3H(p,q) dpdq = e-O'-l. By using Boltzmann's ansatz e = e7)+Ll7) with e7) := ecan and an arbitrary (not necessarily small) variation L1'1](p, q), the constraints, and the inequality L1'1]e Ll 7) + 1 2: L1'1], one may then show that the canonical distribution represents an absolute maximum of entropy. It is worth noticing that in statistical thermodynamics (and in contrast to phenomenological thermodynamics) the entropy is a more fundamental concept than the temperature, which occurs only as a property of special (canonical or equivalent) probability distributions.
35) everywhere. 37) means that there are neither incoming nor outgoing fields 'outside of all charged matter' and for all times. 35), cancels the advanced ones, and imitates the self-interaction j = i responsible for the radiation damping. This remarkable result illustrates the occasional equivalence of quite differently appearing dynamical relations in deterministic theories (as, for example, also those expressed by causal or extremal principles of mechanics). This connection between local and universal descriptions, somewhat reminiscent of the interpretation of inertia according to Mach's principle, may be a warning as much as a stimulation for other situations in physics.
Because of the analogy one may apply the formal solution known from quantum mechanics, er(t) = exp( -iLt)er(O) for aL/at = o. L is hermitian with respect to the , , inner product (g r, er') := J erer dp dq (that is (e r, Ler) = (Le r, er)), as can again be shown by partial integration. As is known from the Schrodinger equation, this means that the Liouville equation conserves these inner products. 27) since the Liouville operator applied to a function f(e r) satisfies the same Leibniz chain rule Lf(er) = (df/der)Ler as the time derivative.