x4 統計力学の基礎 - phys.sci.hokudai.ac.jp

1 4 . Clausius . d Q. dS . T. S T Q .. [1] . p p .. p = 1. (1).. (i) S p . f (p) .. S = p f (p ). (2).. (2) f (p ) S . (ii) . j = 1, 2 S ( j) 1 + 2 S (1+2) .. S (1+2) = S (1) + S (2) . (3). (iii) j = 1, 2 . 1 p(1) 2 p . (2). 1 + 2 ( , ) p . (1+2).. p(1+2). = p(1). p . (2). (4). (i)-(iii) f (p) (3) (2) (4) .. 0 = S (1+2) S (1) S (2).. = p f (p p ) . p(1) (2) (1) (2). f (p ) . p(1) (1). p(2) (2). f (p ).. (1) p(2). (= 1) p(1). (= 1) .. = p(1) (2) (1) (2). p f (p p ) p(1) (1). f (p ) p(2). p(1). p f (p(2). (2). ).. (2) [ (2) ]. = p(1) (1) (2). p f (p p ) f (p(1).)

2 F (p ) .. 1. p(1) (2). p . f (pq) = f (p) + f (q). (5). f (p) f (p) . (5) q . p f (pq) = f (q). q = 1 . k f (p) = , k f (1) . p p . f (p) = k ln p + . (6). f (p) . (6) (2) S .. S = k p ln p + p = k p ln p + .. lim S = 0 0 . T 0. T = 0 = 0 p = 0 . lim p ln p = lim p ln p = 0 lim p ln p = 0 lim S = 0 . p 1 p 0 T 0 T 0.. 0 .. S = k p ln p . (7).. k = 10 23 J/K (8). Boltzmann (7) .. Boltzmann 1872 (i) Boltzmann (ii) . (iii) (7) (H ) (7). Boltzmann . Gibbs von Neumann . (7) Shannon .. 2. [2] .. (7) . 3 p .. [M] (microcanonical ensemble). V . U N . (U, V, N) W = W(U, V, N).

3 (1) (7) .. S S + p 1 = k p ln p + p 1 .. S p 0 . S . 0= = k(ln p + 1) + p = e /k 1 = . p . p . ( ) W . 1. p = . (9a). W. (7) (thermodynamic equilibrium) . S eq S [p = 1/W] . W. 1 1 1. S eq = k ln = k ln = k ln W. =1. W W W.. S eq = k ln W (9b). (9b) Einstein Boltzmann (Boltzmann's principle) . W = eS eq /k .. 1. p = ( ) (10a). W. W(U, V, N) (10b). S = kB ln W ( ) (10c). W = W(U, V, N) . (10c) T . 3. P dS = (1/T )dU + (P/T )dV ( /T )dN . ( ) ( ) ( ). S 1 S P S . = , = , = (11). U V,N T V U,N T N U,V T.. S / U > 0 . T 0 .. [C] (canonical ensemble). V . E . U.

4 U= p E (12).. (7) (1) (12) . T Lagrange .. F U T S + p 1 = p (E + kT ln p + ) . (13).. F Helmholtz F . p 0 . F. 0= = E + kT (ln p + 1) + p = e (E + )/kT 1 e E /kT . p . p e E /kT Gibbs (1). p . e E /kT . p = , Z= e E /kT . (14). Z . Z p = e E /kT /Z (13) . Helmholtz Feq F[p = e E /kT /Z] . [ (. 1 E . )] . 1 1. Feq = p E + kT ln = p kT ln = kT ln Z = eFeq /kT .. Z kT . Z Z.. ( = 1/kB T ). e E . p = (15a). Z . Z(T, V, N) = e E (15b).. 1. F = ln Z (15c).. 4. Z = Z(T, V, N) . (15c) . S P dF = S dT PdV + dN . ( ) ( ) ( ). F F F. = S , = P, = (16). T V,N V T,N N T,V.

5 U (12) (15a) (15b) . 1 E . U= e E = ln Z (17). Z . (17) .. ( ). U 1 2 E 2 E 2. CV := = ln Z = (18). T V,N kB T 2 2 kB T 2. (17) (18) .. [G] (grand canonical ensemble).. E N . N .. N= p N (19).. (7) (1) (12) (19) . T , , Lagrange .. U T S N + p 1 = p (E + kT ln p N + ) . (20).. p 0 .. 0= = E + kT (ln p + 1) N + . p . p . p = e (E N + )/kT 1 e (E N )/kT . p e (E N )/kT (1) . p . e (E N )/kT . p = , ZG = e (E N )/kT . (21). ZG . 5. Z G p = e (E N )/kT /ZG (20) . eq [p = e (E N )/kT /ZG ] .. [ (. 1 E N . ) ] . 1. eq = p E + kT ln N = p kT ln = kT ln ZG.. ZG kT.

6 ZG. 1. = e eq /kT . ZG.. ( = 1/kB T ). e (E N ). p = (22a). ZG . ZG (T, V, ) = e (E N ) (22b).. 1. = ln ZG (22c).. ZG = ZG (T, V, ). (22c) . S P N d = S dT PdV Nd . ( ) ( ) ( ).. = S , = P, = N (23). T V, V T, T,V. U (12) (22a) (22b) .. 1 (E N ) . U= e (E N + N ) = ln ZG + N (24). ZG . (23) N . 2N := N 2 N 2 . 1 2. N 2 N 2 = ln ZG . (25). 2 2.. [3] .. 6.. N g(E ) .. 1 . N. d3 r j d3 p j g := g(E ) g := g(H) (26).. N! i=1 (2 )3. (r j , p j ) j .. N 3 3 3 3 3 3. d3 r j d3 p j d r1 d p1 d r2 d p2 d rN d pN. 3. = 3 3. (27). i=1. (2 ) (2 ) (2 ) (2 )3. H .. N. p2j.

7 N . N. H= + V(|ri r j |). j=1. 2m i=1 j=i+1. m p2j := p2jx + p2jy + p2jz V .. (26) 2 . p x 2 = 10 34 J/s N! N .. N! .. 7.