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N

f

p

q

q

p

Rf

f

p

q

Rf

i

pi

H(q, p)

pidpidt

=H

qi;

qi

dqidt

= H

pi,

i [1, f]

(q, p)

M,

(q, p)

fi=1dqidpi (q, p) (q, p)

fi=1dqidpi

(q, p)

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(q,p,t)

ddt

= 0.

t

+f

i=1

qi

qi+ pi

pi

= 0

)=

t+

fi=1

qi

Hpi

+ pi

Hqi

= 0.

(p,q,t)t

= 0

s.s =(H(q, p))

N V E

N

V

T

V

T

N V E

(N , V , E )

N V E = constant N V E(q, p) 1

,

(N , V , E )

1/(N , V , E )

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N V T

T

E

NV T(q, p)

exp(H(q, p))

= (kBT)1

T

kB

N

V

T

ZN(V, T) =

dq

dp exp(H(q, p))

N V T(q, p) = exp (H(q, p)) /Z

V T

E H

V T(q,p,N) exp((HN(q, p) N))

V T =

N

dq

dp exp((HN(q, p) N)) =

N=0eNZN(V, T)

|(ra, rb)|

2 = |(rb, ra)|2

(ra, rb) = (rb, ra)

(ra, rb) = (rb, ra)

(r1, r2 = r1) = 0

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r

nr r r

r

r

nr =N;

r

nrr =E.

{n1, n2, n3, }

nr = 0 1 nr = 0, 1, 2, 3, 4,

P({n1, n2, }) exp

r

nrr

r

nr

=

r

exp[nr(r )] .

P({n1, n2, }) =

r

pr(nr); pr(nr) exp[nr(r )]

nr pr(nr) = 1,

pr(nr) =exp[nr(r )]

S ,

S=

nr

exp[nr(r )] ,

nr = {0, 1} S

SFD, r = 1 + exp [(r )] = 1 +xr xr = exp [(r )]

nr = {0, 1, 2, 3, } S

SBE, r = 1 + xr+ x2r+ x3r+ = 11xr .

SFDBE , r = (1 xr)1

F D

r

pFDBE

, r = xnrr

(1 xr)1 , xr = exp [(r )] .

nr =

nrnrpr =

0x0r+1x

1r

(1+xr)

, nr = {0, 1}

0x0r+1x1r +mx

mr +

(1xr)1 , nr = {0, 1, 2, 3, 4

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