Конспект лекций по статистической физике. Коренблит С.Э - 11 стр.

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    2.2      ’¥®à¥¬ ¨ ãà ¢­¥­¨¥ ‹¨ã¢¨««ï
   „«ï ª« áá¨ç¥áª¨å á¨á⥬, ¯®¤ç¨­ïîé¨åáï £ ¬¨«ìâ®­®¢®© ¤¨­ ¬¨ª¥,
¨¬¥¥â ¬¥á⮠⥮६ ‹¨ã¢¨««ï ® á®åà ­¥­¨¨ ¢¥«¨ç¨­ë \§ ­¨¬ ¥¬®£® á¨-
á⥬®© ä §®¢®£® ®¡ê¥¬ ". ãáâì ¢ ¬®¬¥­â ¢à¥¬¥­¨ t = 0 ä §®¢ë¥ â®çª¨
á ª®®à¤¨­ â ¬¨ X 0 = (qi0; p0i )s1, i = 1  s, £¤¥ s { ç¨á«® á⥯¥­¥© ᢮¡®¤ë,
­¥¯à¥à뢭® § ¯®«­ïîâ ä §®¢®¥ ¯à®áâà ­á⢮ á ¯«®â­®áâìî %H (X 0; 0). ‚
ᨫã ãà ¢­¥­¨© ƒ ¬¨«ìâ®­ , qit ¨ pti ïîâáï äã­ªæ¨ï¬¨ t ¨ ­ ç «ì­ëå
¤ ­­ëå qk0, p0k , ¯®í⮬ã, ¨§¬¥­¥­¨¥ ¢® ¢à¥¬¥­¨ ¬¥àë ¨­â¥£à¨à®¢ ­¨ï, ¢
ᮮ⢥âá⢨¨ á (1.6){(1.8), ®¯à¥¤¥«ï¥âáï 类¡¨ ­®¬ Dt = det jj@xti=@x0j jj
¯à¥®¡à §®¢ ­¨ï Gct (1.8) ®â X 0 !ª X t, â.¥., ®â x0j !ª xtj , ¤«ï j = 1  2s:
     2s                                          t     d 2s X t      @   (X t) @ (xt1; :::; xt2s )
   d X  dx1 dx2 : : : dx2s ; D  d2s X 0 = @ (X 0)  @ (x0; :::; x0 ) : (1.14)
                                                                                              1          2s
„¨ää¥à¥­æ¨àãï ­¥§ ¢¨áï饥 ®â ¢à¥¬¥­¨ ãá«®¢¨¥ ­®à¬¨à®¢ª¨ (1.12), ¨¬¥-
¥¬, çâ® 8 t:
           d  Z            t           t   2s  0
                                                     Z 2s 0 " t d                      t                    t     d   t
                                                                                                                        #
   0=
          dt fX g%H (X ; t) D d X =fX gd X D dt %H (X ; t) + %H (X ; t) dt D ;
                       d                !             d                       !
   ®âªã¤ : dt D = 0 () dt %H (X ; t) = 0 :
                               t                                 t                                                 (1.15)
’.¥. ¨­¢ ਠ­â­®áâì (­¥¨§¬¥­­®áâì) ¬¥àë ¨­â¥£à¨à®¢ ­¨ï ®ª §ë¢ ¥âáï
¥áâ¥á⢥­­ë¬ ®¡à §®¬ íª¢¨¢ «¥­â­ ­¥á¦¨¬ ¥¬®á⨠\ä §®¢®© ¦¨¤ª®áâ¨":
                 d % (X t; t)  @%H (X t; t) + X                 2s @%H (X t; t)
                 dt    H
                                                  @t            i=1
                                                                     _ ti
                                                                     x        @xti =) 0:                           (1.16)
„®ª ¦¥¬, ­ ¯à¨¬¥à, ¯¥à¢®¥ ¨§ à ¢¥­á⢠(1.15). ®áª®«ìªã ¤¨ää¥à¥­æ¨-
஢ ­¨¥ ®¯à¥¤¥«¨â¥«ï (1.14) ᢮¤¨âáï ª á㬬¥ ®¯à¥¤¥«¨â¥«¥© á ¯à®¨§¢®¤-
­ë¬¨ ®â ª ¦¤®£® á⮫¡æ (¨«¨ áâப¨), ᮮ⢥âáâ¢ãîé¨å ¤¨ää¥à¥­æ¨-
஢ ­¨î ®â¤¥«ì­ëå xtj , â® ¤«ï ¯à®¨§¢®¤­®© ®â ¥£® «®£ à¨ä¬ ¨¬¥¥¬:
    d ln Dt = 1 dDt = @ (x01; :::; x02s ) X                  2s @ (xt1 ; :::; x_ tj ; :::; xt2s )
        dt         Dt dt @ (xt1; :::; xt2s ) j=1 @ (x01; :::; x02s ) =
         2s @ (xt1 ; :::; x_ tj ; :::; xt2s )      2s @ x_ tj
                                                                      0
                                                                   s @ q_t @ p_t
                                                                                           1
        X                                         X               X
    =                                                               @ it + ti A =)                               (1.17)
        j=1 @ (x1 ; :::; xj ; :::; x2s ) j=1 @xj i=1 @qi @pi
                 t           t           t                 t
              0 2                      2H 1
           X
    =) @ t t
            s      @  H              @        A  = )   0 ;    ¤«ï   ­¥¯à¥à뢭ëå                 @ q_it ; @ p_ti ; (1.18)
            i=1  @q @p @pt @qt
                       i    i         i    i                                                    @qt @pt
                                                                                                i       i