Thermodynamics of a pion gas

Volume 61A, number 5

PHYSICS LETTERS

30 May 1977

THERMODYNAMICS OF A PION GAS David E. MILLER Department of Theoretical Physics, University of Bielefeld, University of Bielefeld, 48 Bielefeld, F.R. Germany Received 24 March 1977 The thermodynamics is presented for a gas of uncharged pions coupled to a static nucleon background through a classical Yukawa interaction. By using invariant phase space in the grand canonical ensemble it is possible to write a general analytical form for the thermodynamical functions.

The development of a relativistically covariant statistical mechanics [1] involves an eight dimensional invariant phase space volume ~1associated with each of the particle coordinates and momenta. It has been shown how the correct thermodynamics for the ideal relativistic quantum gases [2] is retrieved from the proper physical assumptions on the partition function which reduces this relativistic phase space d ~l in the rest frame to the usual non-relativistic form gV(2 ir 2)1d3p whereg is a phase space weight, Vis a macroscopic volume and p is the particle momentum (with c = = 1). This result readily yields the thermodynamics of the classical ideal relativistic gas [3] as well as the correct Stefan-Boltzmann law in the high temperature limit. We shall now consider a thermodynamical model involving a static background of nucleons which emits and abscvrbs pions with a rest mass m for the process governed by the energy spectrum for each particle

At this point we must investigate the properties of the phase s~aceintegral. A change of variable toy 3 yields in = + m /m and the coupling c” = cVm the thermodynamic limit the pressure P(8, A) in the form

H(p)~42 +m2 +c’(p2 +m2)~

P=B~

~ f

flP =

~

21r2 k1 k 1

dy yv’?ii~iexp{—k~m(y+ c”/y2)} (2)

This general type of [5]a of integral is known to be evaluateable in terms series of incomplete gamma functions. However, foritthe case offor theevaluation energy spectrum described by (1), is clearer purposes to write this result in terms of a series in the modified Bessel functions K~(x)and the exponential integrals E~(x), both of which are known and tabulated [6]. As a simplification we choose a variable x for ma and define a constant B asgm4/2ir2. Thus the evaluation ofF in (2) is written as Akrl

(1)

Since in this model the pions are not interacting with each other, we may evaluate the grand partition function as a product over the particle momenta. Furthermore, for this case it is possible to define a relativistic activity (fugacity) A in terms of a properly determined chemical potential p and an inverse temperature as exp{j3p}. Under the proper bounding assumptions on A for the relativistic gas, as will be stated later, we can write the sum over the momenta as a phase space integral and proceed in the usual way [4] to evaluate the relativistic thermodynamical functions.

a

1~_[~—K21cx÷ ~(—kxc”rjE(~~.~)

—

~(2l —3)!! E2~÷2, (kx))] 1=1 (21)!!

(3)

where the symbol (n)!! means the product of every integer up to n with (—1)!! = 1. Similarly we find for the average particle density ~ as ~

mB ~Ak[_i~_K2(kx)+ k= 1 —

~

~

(kx)

Lkx

(2!— 3)!!E(~))]

(4)

1=1

291

Volume 61A, number 5

PHYSICS LETTERS

and average density ë~as B ~ AK

~=

k= 1

+ ~ n1

1~~3

provides a basis for further research in this direction. Here we have investigated a simple thermodynamical

K

2(kx) + K1 (kx)

(—kxc”)’~[E

L

~

2n— 1(kx)

—

nE2~(kx)

(5)

_________ _____

—

1=1

(2!)!!

30 May 1977

2n+21—

—

flE2n_2l(kX))])

We now investigate briefly the structure of these thermodynamical functions. It is now quickly shown by using the asymptotic structure [6] of E~(x)that for the limit x 00 the first correction always vanishes for all powers of c”. Thus for small coupling c” the nonrelativistic limit yields only the ideal gas term. The ultrarelativistic limit of x 0 and A 1 readily yields blackbody radiation in the leading term. The corrections to the ideal relativistic gas are most significant for ternperatures giving thermal energies approximating the mass energy where we find a coupling dependent correction. The structure of the energy spectrum (1) demands for the convergence of (2) that A be bounded by exp{x(l + c”)}. ForA at this point we must insert an additional term before replacing the sum over states by an integral. This procedure relates to the familiar process of Bose-Einstein condensation [4], in the ideal Bose gas. Such an extended range for the relativistic activity has been related by Montvay and Satz [7] to a clustering effect in the ideal Bose gas giving rise to a critical temperature. The known phenomenon of pion condensation in nuclear matter [8] as well as the study of the symmetry behavior in gauge theories [9]

model for a noninteracting pion gas as a correction to the ideal relativistic Bose gas. In this model the nucleons in no way enter directly into the dynamics but only act as an external source or sink for the pions governed by the activity. These great simplifications allow the treatment of the pionic system through a highly simplified energy spectrum. Furthermore, we have also not distinguished between the charged and the neutral pions. These drastic assumptions allow the thermodynamical treatment of a system which is generally handled by using field theoretical methods even for the static model [10].

—~

—~

292

—~

The author would like to thank P. Frd, F. Jegerlehner, J. Karczmarczuk, R. Page and especially H. Satz for many discussions on this problem. References [1] B. Touschek, Nuovo Cimento, 58B (1968)

295.

[2] M. Chaichian, R. Hagedorn and M. Hayashi, Nuci. Phys. B92 (1975) 445. [3] F. Jüttner, Ann. d. Phys. (Leipzig) 34 (1911) 856. [4] K. Huang, Statistical Mechanics (John Wiley, New York, 1963). [5] D.E. Miller, Bielefeld Preprint, BI-TP 76/21, 1976. [6] M. Abramowitz and l.A. Stegun, Handbook of mathematical functions (Dover, New York, 1965). [7] I. Montvay and H. Satz, Bielefeld Preprint, Bl-TP 76/17, 1976.

[8] D.A. G.E. Kirzhnits Brown andand W. A.D. Weise,Linde, Phys.Ann. Reports 27C (1976) [9] Phys. (N.Y.), 1011. (1976) 195. [10] E.M. Henley and W. Thirring, Elementary quantum field theory (McGraw-Hill, New York, 1962).