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Legendre Transformations

y,x. p, f. y. p. x. f. Legendre Transforms. Legendre Transformations. Consider y=y(x) with a slope p=p(x). f (p)= y – p x. f (p) is the Legendre transform, where y = y(x), x = x(p) and p is the independent variable. Natural Variables. Q. E 1. E 2. const. S’(E, X , c). X.

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Legendre Transformations

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  1. y,x p,f y p x f Legendre Transforms Legendre Transformations Consider y=y(x) with a slope p=p(x) f(p)= y – p x f(p) is the Legendre transform, where y = y(x), x = x(p) and p is the independent variable

  2. Natural Variables

  3. Q E1 E2 const. S’(E,X, c) X S(E,X) E Variational Statement for S ET doesn’t change, E1,E2 changes (heat exchange) 1)Add an int. constraint  move out of the equilibrium E,X plane (reversibly),  S’(E,X; int.constr) 2)Add an adiabatic wall 3) Remove int.constr.  move back into E,X plane  S > S’  S(E,X,) > S’(E,X;int. constr.) Variational principle: Equil. State is the state with max S(E,X;int. constr.)

  4. Q E(1) E(2) Variational Statement for E

  5. continuation

  6. dE

  7. Goal: Molecular propertiesThermodynamic properties Goal of Stat Mech Start with mechanical properties (P, E, V) can be derived from Q.M. or C.M behavior. Obtain nonmechanical properties (S, G) using thermodynamic relationships Macroscopic only a few parameters to specify (V, [], T,etc) Microscopic for N particles, there are ~10N Q. states, and to obtain macroscopic properties, we need to know in which of the 10N states is the system. STATISTICAL MECHANICS TO THE RESCUE!!!!

  8. Postulate of Stat Mech. Stat. Mech. Postulate: If you can calculate a mechanical property Xi consistent with the macroscopic parameters, then, <Xi> =macroscopic thermodynamic X

  9. Ensembles What is an ensemble? Ensemble= Assembly of all possible microstates (consistent with macroscopic constraints) Virtual collection of a large # of microsystems, each one being an exact macroscopic copy of the system under study. Though all systems are macroscopically identical, they are not equal at the molecular level!

  10. Surface trajectories and ergodicity Presence of constraints  trajectories in state surface Waiting long enough, all possible microscopic states will be visited Ergodic principle: If over time all microstates are visited, then a <time> equivalent to <ensemble>

  11. A 1 2 3 V=AxV N=AxN E=AxE Microcanonical N,V,E There are identical A replicas with same N,V, and E Microcanonical Ensemble There are W(E) Q.M. states which correspond to N,V,E. E is the solution of Schrödinger Eq. Of all possible Ej, there are W(E) degenerate states with energy E Molecular level, there are different states (i.e. diff. Q. number) Thermodynamically, they are all equal with N,V,E parameters

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