A method of finding thermal average by differential operation

~)

0038-1098/9153.00 +.00 Pergamon Press plc

Solid State Communications, Vol. 77, No. 5, pp. 345-346, 1991. Printed in Great Britain.

A METHOD

OF FINDING

THERMAL

AVERAGE

BY

DIFFERENTIAL

OPERATION

Toshio TSUZUKI

Department of Physics, Faculty of Science, Tohoku University, Sendal 980, JAPAN

(Received on 14 November 1990 by H. Kamimura) We propose a practical method of finding the thermal average of a set of operators A under the assumption that we know the Heisenberg equations of motion of A in the interaction representation in a closed form. This is accomplished by the aid of the coherent state representation for the bose field and by the aid of the Grassmann number representation for the fermion field. The problem to be solved is to find out the expression of A in the respective representations by taking full account of the quantum zero-point fluctuation only.

come through at(t) = at" exp[-iett] and a+(t) = a +. exp[+iett], and through external field if applied.

This report is composed of two parts. In the first part we show that the following thermal average is calculated by differential operation by the aid of the formula that < a > - T~b(/~)A(a+,

_- exp

Let us come back to derive (1). We use the following formula [1,2,4,5]:

a)

.f (et) ~-~k-k"b-~

~=~,=o'(1)

= / d~*d~dr/*dr/exp[-~t{~;(~t where ~(/3) = exp[-~H0]/Z(~) and Z(/3 ) = Trexp [-/~H0]- The Hamiltonian ~r0 = ~--~tetat+at describes free motions of bosons (fermions) with annihilation and creation operators at and at+ of quantum number(s) k, and f(e) = 1/(e/~'q:l) the bosonic or fermionic distribution function with temperature 1/#. Operator A(a+,a) is a normal-ordered form composed of arbitrary numbers of operators {at} and {at+}. For the bosonic ease ~t is a complex number which specifies the Glauber coherent state of boson at [1,2] and for the fermionic case ~t is n Grassmann number with antlcommuting algebra which specifies the coherent state of fermion at [3,4,5]. So our problem is to find the coherent state representation of A, A(~*, ~).

• p(C, 7). A(~*, ±~),

(3)

where d~*d/~ = IIk{d~;d~t/l¢ } for the boson and the same definition without ~r for the fermion. Take the upper signs for the boson and the lower ones for the fermion. The latter implies the antiperiodic boundary for the fermion. Quantity A(~*,17) is obtained by the replacement {a +} --, {~} and {at} ---* {~t} in /]( a+, a). If we notice

1[

1

p(~*,¢/) = Z-----~exp -E(l:Fe-/~e')~;tlt , k J

In the second part we discuss A(~*,~) under the assumption that the equation of motion for A is given in the interaction representation with respect to H0 by

(4)

with zC~ ) = IIt(1 :F e-~C') ~:t, it is a straightforward task to find C1). A few comments are in order. Ca) We can compute the thermal average < A > by the differential operation if we know the coherent state representation AC~*, ~/) irrespective of statistics. (b) Since the differential operation is defined by an exponential operator, we can rewrite (1) as

A(a+,a;t) = AoCa+, a;t)

- r/t) -t- r/;(r/t :F ~t)}]

+ f' dt'L(a+,a;t,t'). A(a+,a;t'),



(9) where/1o and L are normal-ordered operators of known functional form. The time-dependences of -40 and

o~k jj I~=~.=o (5) 345

346

A METHOD OF FINDING THEP~IAL AVERAGE BY DIFFERENTIAL OPERATION

A(~',~;t)

(c) As an extension of (1) for u product of two normalordered operutors Al(a+,a) and A2(a +, a) we find

= Ao(~*,~;t)+ <

/'

Vol. 77,

dt'L(~*,~+V*;t,t')A(C,~;r). (8)

~1" A~ >

Ir

----exp _

L

k

f(ek)-~-~ -E-~ AI(~*,~ -I- V*)A2(~*,~) °~kJ

No: 5

~=~'=0' (6)

(7) where V* = 0/0~*. We can obtain this type of formula for s product of am arbitrary numbers of normalordered operators. Next let us discuss how to find the time-evolution of thermal average < A(t) >. The equation of motion (2) is expressed as

Note that this equation contains the influence of quantum serc~point fluctuations completely. So the present method allows us to calculate < / ] ( t ) > by successive two steps : First examine the dynamical effect on /] in the presence of the sero-point fluctuation only by solving (8), then put the thermodynamic fluctuation to A(~*,~;t) by the aid of (1). We can compute the effects of zero-point and thermodynamic fluctrstions in the automatically equitable treatment, even if we introduce an approximation to solve (8). It is easy to extend the present method to the case thut A and are vector and tensor operators respectively. The present method has been applied to the case of bosonic field [6,7]. We have studied the partition function [0] and the spin dynamics [7] of a spin-boson system. The author expects this method to be practically useful for various problems.

Acknowledgementt- This work is supported in part by a Grant-in-Aid for Fundamental Scientific Research from the Ministry of Education, Science and Culture under Grant No. 02640269.

References 1. R.J. Glauber, Phys. Rev. 131 (1963), 2766.

4. Y. Ohnuki and T. Kashiwa, Prog. Theor. Phys. e0 (1978), 548.

2. P. Csrruthers and M.M. Nieto, Rev. Mod. Phys. 40 (1968), 411.

5. D.E. Soper, Phys. Rev. D18 (1978), 4590.

3. F.A. Berezin, The Method of Second Qnantiza. tion (Academic Press, 1966).

6. T. Tsuzuki, Prog. Theor. Phys. 85 (1991), No.1. 7. T. Tsuzuki, Prog. Theor. Phys. (submitted).