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.. S=k ln(w) Z=exp(-H/kT) d F = - kT ln(Z) dF = - S dT- p dV+ dN - 2010 - , .. - 2010 .. . . : , 2010. 99 .

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- , : 140400 ( ), 20020101 ( ), 20020104 ( ), 20020107 ( ) ( 3 9 2010 ) 2009 , 12 , . - , 20092018 .

- , , 2010 .. , 2010 H T p P M E V m T p N s = 3N 6N = 2s ri pi = m = mvi m i i = 1, 2,..., N qi pj j = 1, 2,..., 3N qj q pj p H 3N pj H(q, p) = + U(q), 2m j=U(q) H H =, = -.

j j pj qj x1 = qp1 U = = ; = = F1x;

1 m xp1 = m ; m = F1x 1 q p q(t) p(t) (0) qj(0) = qj pj(0) = p(0).

j N N T V N N 6N qj pj q, p (q, p) (q) (p) 3N 3N dq dp = dqjdpj j= dqdp dt dt/ dqdp dqdp q p dw dq dp dt - dw = (q, p) dq dp.

(q, p)dq dp = 1.

(q, p) dq dp N q p A = A(q, p) P(p) = pi i= N M(q, p) = ri pi i= q(t) p(t) A(q(t), p(t)) A A A = A(q(t), p(t))dt - A(q, p)dw = A(q, p)(q, p) dq dp.

6N A A A A A tn = t n (n = 0, 1, 2,..., M - 1).

M dqdp Mdqdp M Mdqdp = dqdp = dw M dqdp M M (q, p) A A(q, p)M(q, p)dqdp = A(q, p)(q, p)dqdp = A.

M(q, p)dqdp A (q, p) 1(q(1), p(1)) 2(q(2), p(2)) (q, p) = 1(q(1), p(1)) 2(q(2), p(2)) dq(1) dp(1) dw1 = (q, p) dq dp = 1(q(1), p(1)) dq(1) dp(1) 2(q(2), p(2)) dq(2) dp(2) = (q,p)2 (q,p)= 1(q(1), p(1)) dq(1) dp(2), (q, p) (q, p) = (q( ), p( )) A(q, p) A(q, p) = A1(q(1), p(1)) A2(q(2), p(2)).

A(q, p) A = (q, p)A(q, p) dq dp = 1(q(1), p(1))A1(q(1), p(1)) dq(1) dp(2) 2(q(2), p(2))A2(q(2), p(2)) dq(2) dp(2) = A1 A2, A(q, p) (q, p) q p A A(q, p) = A(q, p) - A.

A(q, p) DA = (A)2 = A2 - 2A A + A2 = A2 - A.

A = DA A a =.

A n = 1, 2,..., n n A(q, p) = A (q, p ), = A (q, p ) n A = A = n a, = A = a DA = (A - A)2 = ( (A - a))2 = = (A - a)(A - a) = (A - a)2 = nb2.

= (A - a)(A - a) = (A - a)(A - a) = (a - a)(a - a) = 0, (A - a)2 = b nb2 b 1 b No 1 A = = = , A, na a n a N N No = N/n N 1022 A 10- N M (q, p) 6N M(q, p) = (t, q(t), p(t)).

+ (v) = 0, t (r(t), t) v(r(t), t) t r(t) M (q, p) 6N 6N qj, pj 6N, j j 3N (M) + (M ) + (M ) = j j t qj pj j= M 3N 3N j j + + + + = 0.

t qj j pj j qj pj j=1 j= 2H 2H j j + = - = qj pj qjpj pjqj 3N d + + = = t qj j pj j dt j=j j + [, H] = 0, t 3N H H [, H] = qj pj pj qj j= H qp M qp (M) = M.

M qp M = t [, H] = 0.

f f = f(t, q(t), p(t)).

3N df f f f f = + + = + [f, H].

dt t qj j pj j t j= f df = 0, dt f = t [f, H] = 0.

= (q, p) ln (q, p) = ln (q( ), p( )) ln E = H(q, p) P M E = H(q, p) = (H(q, p)).

(H(q, p)) h3N 3N dqdp dqdp.

h3N N! N 1/N! N! dqdp (H(q, p))d = 1, d =, h3NN! (H(q, p)) (q, p) (E ) H(q, p) = E (E ) d(E ) = dE g(E )dE E H(q, p) = E H(q, p) = E + dE E (E )g(E )dE = 1, g(E )dE = d E H(q,p)E +dE dE w(E ) = (E )g(E ) E E w(E ) w(E ) E E = H(q, p) E w(E) = (E)g(E) (E)g(E)E = 1.

E W W = g(E)E = -1(E) = -1(H(q, p)).

S = k ln W = -k ln (E) = -k ln (H(q, p)) k k = 1, 38 10-23 K- S S W = exp.

k S S = -k ln (E) = -k ln (E ) = -k ln (E ) = S.

E V N H E q p H(q, p, V, N) = E E, V, N q p mc(H(q, p, V, N)) q p H(q, p, V, N) = E E 6N H(q, p, V, N) = E 6N - mcH(q, p, V, N) mc(H(q, p, V, N)) = (E - H(q, p, V, N)).

c (x) c x = (x) = (x)dx = 1.

x = c (E - H(q, p))d = 1, c 1 (E - E )g(E, V, N)dE = g(E, V, N) = 1, c = g(E, V, N).

c c h3N E W (E, V, N) = g(E, V, N)E.

S(E, V, N) = k ln W (E, V, N) = k ln(g(E, V, N)E), 1 S(E, V, N) g(E, V, N) = exp( ).

E k dE = T dS - pdV + dN 1 p dS = dE + dV - dN.

T T T S 1 S p S =, =, = -, F = E-T S E T V T N T V,N E,N E,V T p F E T = T (E, V, N) p = p(T, V, N) H(q, p, V, N) g(E, V, N) E W (E, V, N) S (E, V, N) T, p, N H(q, p) = Hi(, ), i i i= Hi(, ) i i Hi(, ) = /2m + W ( ), i i i i W ( ) V i L L3 = V V / i W ( ) = i V i g(E, V, N) W (E, V, N) (E, V, N) H(q, p, V, N) = E (E, V, N) = d.



H(q,p,V,N)E 2mE 3N p2 = 2mE, j j= L (E, V, N) = dp dq = 3N h3NN! p22mE 0qjL j=1 j 1 (2mE)3N/2 (2m)3N/N = 3N/2 L3N = E3N/2V.

h3NN! (3N/2 + 1) h3NN!(3N/2 + 1) n/ n R Vn = Rn (n/2 + 1) E3/2V 3 4m S(E, V, N) = kN ln + ln +.

N5/2 2 3h2 x N ln (x) x ln x - x, ln N! N ln N - N.

1 3 kN p kN S =, =, = - + k T 2 E T V T N 3 E3/2V 3 4m E = NkT, pV = NkT, = -kT ln + ln.

2 N5/2 2 3h T T V N H(q, p, V, N) = H1(q1, p1, V1, N1) + H2(q2, p2, V2, N2), E = E1 + E2, E1 << E2 E, N = N1 + N2, V = V1 + V2,, mc(q, p, V, N) = (H1(q1, p1, V1, N1) + H2(q2, p2, V2, N2) - E).

C 1(q1, p1, V1, N1) = mc(q, p, V, N)d2.

6N2 N 1(q1, p1, V1, N1) = (H1(q1, p1, V1, N1) + E2 - E)g2(E2)dE2 = C 1 1 S2(E - H1(q1, p1, V1, N1)) = g(E - H1(q1, p1, V1, N1)) = exp( ).

C CE2 k H1 << E S2 S2 S2(E - H1) = S2(E) - H1(q1, p1, V1, N1) + , =.

E E T 1 S2(E) H1(q1, p1, V1, N1) 1 H1(q1, p1, V1, N1) 1 = exp exp - = exp -, CE2 k kT Z kT 1 S2(E) exp CE2 k Z T V1 N 1 H(q, p, V, N) c(q, p, T, V, N) = exp -.

Z kT Z = Z(T, V, N) H(q, p, V, N) Z(T, V, N) = exp - d.

kT 1 F (T, V, N) = exp, F (T, V, N) = -kT ln(Z(T, V, N)), Z(T, V, N) kT F (T, V, N) F - H(q, p, V, N) F - E c(q, p, T, V, N) = exp, ln c(H(q, p, V, N)) =, kT kT F - E S = -, F = E - T S, T F = F (T, V, N) dF (T, V, N) = T dS - pdV + dN - T dS - SdT = -SdT - pdV + dN F ln Z F ln Z F ln Z S = - = k ln Z + kT, p = - = kT, = = -kT.

T T V V N N E = F + T S ln Z ln Z 2 E = -kT ln Z + T k ln Z + kT = kT.

T T p = p(T, V, N) H = H(q, p, V, N) Z = Z(T, V, N) F = F (T, V, N) S p H(q, p, V, N) Z(T, V, N) = exp - d = kT 1 p2 W (ri) i = exp - dpi exp - dri.

h3NN! 2mkT kT i ri V (2mkT )1/ exp(-x2)dx = (/)1/2.

N 1 (2mkT )3/2 eV N Z(T, V, N) = (2mkT )3N/2V =.

h3NN! h3 N 3/3/T V 2mk F = -NkT ln + ln + N hF 3 NkT F S = - + kN, p =, = + kT, T 2 V N 3/3/T V 2mk S = Nk ln + ln +, N h2 pV = NkT, 3/3/T V 2mk = -kT ln + ln.

N h3 E = NkT, C = Nk.

2 2E T = 3Nk S S T V T q p E = E1 + E2, E1 << E2 E; N = N1 + N2, N1 << N2 N, V = V1 + V2.

T E(q, p, V, N) = (H1(q1, p1, V1, N1) + H2(q2, p2, V2, N2) - E)N +N2,N C 1 = E(q, p, V, N)d2 = N = N +N2,N (H1(q1, p1, V1, N1) + E2 - E)g2(E2, N2)dE2 = C N 1 1 S2(E - H1, N - N1) = g2(E - H1, N - N1) = exp.

C CE2 k H1 E N1 N S2(E, N) S2(E, N) S2(E - H1, N - N1) = S2(E, N) - H1 - N1 + E N 1 S2(E, N) - H1 + N1.

T T 1(q1, p1) 1 S2(E, N) -Zgc = exp, CE2 k N1 - H1(q1, p1, V1, N1) - 1(q1, p1, N1, T, V1 < ) = Zgc exp.

kT N - H(qN, pN, V, N) - gc(qN, pN, N, T, V, ) = Zgc exp.

kT T V N qN pN gc(qN, pN, N, T, V, )dN = 1.

N= Zgc = Zgc(T, V, H(qN, pN, V, N) N Zgc(T, V, ) = exp exp - dN kT kT N= (T, V, ) Zgc(T, V, ) = exp -, (T, V, ) = -kT ln(Zgc(T, V, )).

kT + N - H(qN, pN, V, N) gc(qN, pN, N, T, V, ) = exp.

kT (T, V, ) + N - E S = -kln(gc) = -k, kT N = N E = H (T, V, ) = E - T S - N = F - N.

d = dF - dN - Nd = -SdT - pdV - Nd.

S = -, p = -, N = -.

T V E = + T S + N F = + N T, V, H = H(qN, pN, V, N) Zgc S p N N N Zgc(T, V, ) = exp (2mkT )3N/2V = kT h3NN! N= N 3/2 3/ 1 2mkT 2mkT = V exp = exp V exp.

h2 kT h2 kT N! N= 3/ 2mkT = -kT V exp.

h2 kT N = - = -, = -NkT ;

kT p = - = -, pV = NkT ;

V V 5 5 S = - = - +, S = kN -.

T T 2 kT 2 kT 5 E = + T S + N = -NkT - NkT - + + N = NkT.

2 kT N N pi H = + Uij, Uij U(rij), rij = |ri - rj|.

2m i-1 i

kT N!h3N kT N (2mkT )3N/2V Zid = h3NN! Z Z = Zid QN, Uij i

kT Uij Uij ij = exp - - 1, exp - = 1 + ij.

kT kT QN rj Jj(r1, r2, , rj-1) = drj(1 + 1j)(1 + 2j) (1 + j-1j) = j- = drj 1 + ij + ijlj + .

i=1 il ij j- Jj (1 + ij)drj = V + 4(j - 1) (r)r2dr.

V i= U (T ) = -4 1 - exp - r2dr.

kT Jj = V 1 - (j - 1), j = 1, 2,..., N.

V N N QN = 1 - (j - 1), Z = Z 1 - (j - 1).

V V j=1 j=F = -kT ln Z = F - kT ln QN.

N V N N N(N - 1) ln QN = ln 1 - (j - 1) - (j - 1) = -.

V V V j=1 j=N(N - 1)kT F = F + (T ) 2V N(N - 1)kT (T ) 3 n - 1 (T ) E = E - = NkT 1 - T.

2V T 2 3V T F N(N - 1)kT NkT (N - 1) p = - = p + (T ) = 1 +.

V 2V V 2V d r < d U(r) = Ur>d(r) kT r > d d u(r) V (T ) = 4 r2dr + 4 (1 - exp - )r2dr = 8v + w, kT kT 0 d v w = U(r)r2dr V d NkT 4v(N - 1) N - p = 1 + + w V V kT a p + (V - b) = NkT, V p 1/V a b NkT 1 + NkT + bp = + .

V V V NkT B(T ) C(T ) p = 1 + + + .

V V V B(T ) C(T ) a B(T ) = b- C(T ) = bNkT a N(N - 1) b = 4(N - 1)v 4Nv, = - w.

V b a/V N(N - 1) N V 3 N(N - 1) = - w, E = NkT + w (w < 0), T kT 2 qp h = h = 1.0510-34 m p2 mE = H = + q2.

2m E q(t) p(t) En = (n + 1/2), n = 0, 1, 2,...





m r I = mr pI E = H = =, p = I.

2 2I p m m = 0, 1, 2,...

2mEm =, m = 0, 1, 2,...

2I m = m r I = mr pp E = H = +, p = I, p = I sin2.

2I 2I sin 2l(l + 1) El =, l = 0, 1, 2,...

2I 2l + Z Mz Mz = m, m = 0, 1, 2,..., l.

h3N 3N N! h3N N h3NN! dqdp dqdp d = h3NN! h3NN! (E) (i, n) i n (E) = (Ei) = gi(Ei) = 1, i,n i gi Ei (E)g(E)dE = 1, g(E)dE dE 1 qmc(, E, V, N) = E,E, qmc(, E, V, N) = (E - E) g(E, V, N) g(E, V, N) W = g(E, V, N)E S(E, V, N) = k ln g(E, V, N)E.

1 E(V, N) qc(, T, V, N) = exp -.

Z kT E(V, N) Ei(V, N) Z(T, V, N) = exp - = g(Ei, V, N) exp -.

kT kT i Z 1 N - E(V, N) qgc(, N, T, V, ) = exp.

Zqgc kT Zqgc = Zqgc(T, V, ) N E(V, N) N Ei(V, N) - kT kT kT kT Zqgc(T, V, ) = e e = e g(Ei, V, N)e.

i N=0 N=N pmj H = H(qj, pj); H(qj, pj) = + qj.

2m j= pe mj N Z = Zo ; Zo = exp - dpj exp - qj dqj = Nh3 2mkT 2kT - 1/e 2kT 2e kT = (2mkT )1/2 =.

Nh3 m2 Nh ln Z E = kT = NkT T E C = = Nk.

T Zo 1/T 1 1 1 E = Ec + Epot = NkT + NkT ; C = Cc + Cpot = Nk + Nk.

2 2 2 nj (n1, n2,..., nN) [nj] N E[n ] = (nj + 1/2) j j= nj N! n1, n2,..., nN N N N! e N N E[n ] e e j N Z = exp = exp - (nj + ) = Zo ;

N kT N kT j=1 nj=[nj] exp e 1 e 2kT Zo = exp - (nj + ) = = N kT 2 N nj=1 - exp kT To exp - exp e e 2T = =, N To N 1 - exp - 1 - exp T To = k T = To 1 1 E = N + = N + 2 To exp - 1 exp - T To 2 exp exp To T C = Nk 2 = Nk T To exp - 1 exp - T C T To 2 1 To To To E = N + exp -, C = Nk exp - ;

2 T T T T To 1 T E = NkTo + NkT, C = Nk.

2 To To 1013 To 103K To To N2 O2 H2 HCl HBr To K 3340 2230 5800 4140 N pj H =.

2I j=p e 2e N 2IkT Z = Zo ; Zo = d e dp = (2IkT )1/2, hN hN 0 ln Z E = kT = NkT, T C = Nk.

kT k I (m1, m2,..., mN) {mj} N 2mj E{m } =.

j 2I j= 2m2 Tr - - m2 - me e e N 2IkT T Z = Zo ; Zo = e = e = e.

N N N m=- m=- m= 2 T Tr 2Ik Tr ln Zo() C = kN 2.

C T Tr 1 Zo m2 m - e e e 2e Zo = e = e dm = ()1/2 = (2IkT )1/2, N N N hN m=m= ln Zo 1 E = kT N NkT, C = Nk, T 2 Zo E C T Tr e Zo (1 + 2e ), hN Tr - ln Zo T E = kT N 2NkTre = 2NkTre, T Tr E Tr T C = = 2Nk e = 2Nk-2e, T T Tr Tr N2 O2 H2 HCl HBr Tr K 2.86 2.07 85.4 15.2 12. L m V = L2 En = En,n2,n3 = (n2 + n2 + n2), (n = (n1, n2, n3), n1, n2, n3 = 1, 2,...).

2mL2 1 2 Z 3 e Ttr e nN Z = zo, zo = exp - n2 = exp -, N T N n1=1 n=2 2 T Ttr = = 2mL2k Ttr L = 1 Ttr 10-15 K T Ttr n2 n2 exp - exp - dn = ()1/ n= N (2mkT )3/2 eV Z h3 N N H = Hi, i= Hi i N Z = Zo Zo 3n n n 3n q1 = X q2 = Y q3 = Z q4 = q5 = q6 = q4 q 3n - 6 3n - 3n U(q) = cjqj.

j=7(6) 1 kT k 2 1 kT k 2 kT k kT 1 kT k 2 kT k n (3n-3)kT (3n - 3)k (3n - )kT 5 (3n - )k C = 2 C = k k 5 k k 2 T < To (el) Ei (el) (el) E(el) = E1 - E0 kT T Tel E(el) Tel k (el) E(el) E0 Tel 200 K (el) (el) E1 E2 = E2 - E(el) (el) (el) E0 Tel E0 E1 Tel - (el) E1T 1 + g1 - kT kT T Zo = g0e + g1e = g0e e, g g0 gTel (el) (el) T E ln Z0 E0 g1 e = kT = kTel + kTel Tel N T E1 gg1 T 1 + e gTel Tel 2 T T C 1 E g1 Tel e g0 Tel e = = k = k N N T g0 T g1 T Tel 2 Tel 2.

1 + g1 - g0 + T T e e g0 g C/N T = Tel {l } = l + l + + l.

i 1 2 N l li i i = exp(i) (q2, q1, q3,..., qN) = (q1, q2, q3,..., qN) (q1, q2, q3,..., qN) = (q2, q1, q3,..., qN) = 2(q1, q2, q3,..., qN), 2 = 1 = = - = (A) = pP l (q1)l (q2)...l (qN) 1 2 N p p P p p = p (S) = P l (q1)l (q2)...l (qN).

1 2 N p (A) l (q1) l (q2)... l (qN) 1 1 l (q1) l (q2)... l (qN) 2 2 (A) =.

l (q1) l (q2)... l (qN) N N N l = l l1 = l1 (A) l1 = l2 = l3 (S) = l l nl + N - E{l },N i = exp ;

kT N = nl, E{l },N = lnl; = l, i l l l l + ( - l)nl l = exp = l,n.

l kT l l + ( - l)nl l,n = exp l kT l nl l,n l l l l nll l l - l l,n = 1, exp exp nl = 1, l kT kT nl nl - l l = -kT ln exp nl.

kT nl l l l + ( - l)nl nl = - = nl exp.

kT nl - l = l = -kT ln exp nl.

kT nl l l T, V, nl = 0, - l - l - l nl kT kT kT e = 1 + e, = -kT ln + e.

nl=0 l - - l l e kT fl(F -D) = - = + 1 = nl l - - - l - i e kT + 1 = gi e kT + 1 = N - = l i = (T, V, N) T, V, T, V, N E = + T S + N = fl(F -D)l, l fl(F -D) = nl T = = F = fl(F -D)(l) fl(F -D) = nl nl = 0, 1, 2,...

- l - l -1 - l nl 1 kT kT kT e = - e, = kT ln - e.

nl=0 l -l l - fl(B-E) = - = exp - 1 = nl kT -l nl l exp 1 l kT -1 - l - i - - = exp - 1 = gi exp - 1 = N kT kT l i = (T, V, N) T, V, T, V, N E = + T S + N = fl(B-E)l, l fl(B-E) = nl fl(B-E)(l) fl(B-E) = nl T = l - exp 1 0 - kT nl kT - l l fl(B-E) fl(F -D) exp = A exp - fl(M-B).

kT kT p2 V l p, p =, dp.

2m hl V p2 h3n N A exp - dp = N, A =, n =.

h3 2mkT (2mkT )3/2 V A A 1 n T m n 1028 T = 300 h = 6.63 10-34 k = 1.38 10-23 / m 10-26 A 10- m -30 A V T F (= ) = 0, N T,V = c c / k k = /c = 2/c = 2/ p = k = n /c n k = = pc - i ni = exp - 1, kT i L exp(iki(i + L)) = exp(ikii), i = x, y, z, exp(ikiL) = 1.

2 2 L ki = ni, ki = dk =, ni = 0, 1,..., , L L ki L dk.

ki nx, ny, nz exp(i(kxx+kyy +kzz)) dk L V 2 4k2dk = 2d = g()d, 2 2c = kc + d V V 2d dN = ng()d =.

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