- Email: [email protected]
On the relativistic partition function of ideal gases
Volume
127B, number
PHYSICS
6
ON THE RELATIVISTIC
PARTITION
LETTERS
FUNCTION
11 August
1983
OF IDEAL GASES
Yu .M, SINYUKOV USSR State Standards Committee, Moscow, USSR Received
3 December
1982
The covariant partition function method for ideal Boltzmann and Bose gases is developed within quantum field theory. This method is a basis to describe the statistical and thermodynamical properties of the gases in canonical, grand canonical and pressure ensembles in an arbitrary inertial system. It is shown that when statistical systems are described relativistically it is very important to take into account the boundary conditions. This is due to the fact that an equilibrium system is not closed mechanically. The results may find application in hadron physics.
I. The requirement that the descriptions of a thermodynamic system in different inertial systems should correspond to one another unambiguously was long considered to be of fundamental rather than practical character. The problems in hadron physics, however, have now confronted us with the practical need of a consistent relativistic formulation of statistical theory. As an example, we can indicate the last series of papers by Hagedorn and collaborators [ 1 ] where the hadron-quark phase transition was investigated largely based on a covariant formulation of statistical mechanics [2 ] . The reason for writing this paper is that the relativistic formulation of statistical mechanics proposed by Touschek [2] and appearing to be generally accepted (see ref. [3] ) cannot be regarded as physically correct. What we mean is as the following. The Boltzmann equation S = k In r and the invariance of entropy S [4] result in the requirement that the phase volume r‘ = J dF is invariant. To guarantee this, Touschek proposed [2] that the element of phase volume, dI’, should also be invariant. Then a relativistic expression for the phase volume occupied by a gas of N particles with mass m in a canonical ensemble (referred to as the partition function 2) is
where up is the four-dimensional velocity of a thermodynamic system, that moves as a whole (with velocity u); the asterisk denotes the quantities referring to a system at rest. The assumption that the dr is invariant looks natural and is the only means of securing invariance of JdI’. This means, in fact, that all the states of the ideal gas transform according to the usual Lorentz transformation for a totality of free particles. And precisely this assumption that the element dr of a phase volume occupied by an ideal gas is invariant, is incorrect. An ideal gas in a state of global equilibrium should be considered in a container (“box”) of volume V. The phase volume element dr of a system is not invariant because the particles in a box are not free, but influenced by the walls of the container where they happen to be. A nontrivial action of the walls confining the system was clearly manifested in relativistic thermodynamics by Callen and Horwirz in ref. [S] A thermodynamical system in a box is not closed; therefore its energy and momentum no longer form a four-vector. The energy and momentum transformation law for these systems are [6] (9 is the pressure): E = Un(E* + &V*),
exp[-(u’Pi)/T]6(pi2-m2)d4pi, (1) 0 031-9163/83/0000-0000/$03.00
0 1983 North-Holland
P = U(E, + TV*).
(2)
The formulas (2) cannot be explained within the covariant formulation of statistical mechanics [2] , be443
Volume
127B, number
6
PHYSICS
cause in the framework of that approach an ideal equilibrium gas in volume Vis incorrectly regarded as a closed system. We say that the representation (1) for the partition function is not related to a description of a statistical system by a moving observer: the integrand does not correspond to the momentum distribution function in a moving reference frame. 2. Consider a system of free uncharged scalar particles of mass m. in a rectangular box of dimensions a X b X c. The equation for a field and the condition for confining the field in a box are :
(0 -
m&w =0,
q(x) = 0
(3)
at the boundary
and outside the box.
(4)
We note that unlike the condition of “impenetrability” of the walls (4), imposing periodic boundary conditions on the field is not a Lorentzcovariant operation. To construct the basis of the functions we make use of the Lorentz-invariant scalar product
(5) Expanding the arbitrary solution of eqs. (3), (4) on the corresponding complete orthonormalized set of solutions, we have after quantization
Here a:, sol are the annihilation and creation operators of states with quantum numbers (Y. Our task is to study the energy and momentum distribution functions of particles in an arbitrary reference frame. For this purpose we expand the basis functions q(y in plane waves normalized to one particle in a box:
LETTERS
11 August
4&n=u eklm 0
1983
+ kklm- u,
Pklm = UEklm + UOkklm’
k klm = (n-k/a, d/b, nmlc),
eklm = (rnt + k&,)‘/2, (8)
The eight operators aP, are related to a, by “p, = aklm =
i2~312(klm/Iklm~)[p~~,/uO(~~pk~m)] l12aKLM
(aKLM = aor) .
(9
The commutation relations for these operators follow from (6). These commutators are not the standard Bose-commutators because the set {+, } are not the basis functions for eqs. (3), (4). Howe&, the introduction of the operators up is a useful trick. It enables one to express the stagstical quantities in energy-momentum variables. We note in the first place that the expansion (7) is the expansion of wavefunction of the system on the sum of orthonormalized states corresponding at Y113 5 A/lp, 1to particles with momentum poi. The coefficients multiplying aKM in (9) are the coefficients of the expansion of q, on pP . The square of the absolute value of these coeffic&ts is the probability W@,) of finding a particle with momentum p, if the system is in a state (Y.The operator
that acts on states . . . (cz~)~ .. . 10)thus corresponds to the average number of particles with momentum par. From (9) follows the explicit dependence of the operator Nia on the velocity of a system. Using (S), (9), we find that the normalization of the state Ip,) E aiq 10)is the same as the probability of observing a particle in this state (p, Ip,) = WN;ala) = 1
p:
= w&J>
8uowP,)
pppp,=
(2~:abc/uo)-1i2
exp (-ip;x).
(7)
For every level (Y= {K,L,M}, (K,L,M = 0,l . ..) the momentum p, z (pklrn) takes on 8 values correspondingtok=+K,l=+L,m=+M,
444
c W(p,) = 1 PQ!
(a fixed).
(10)
This allows us to write down the contribution to the mean value of some linear operator B from a oneparticle state la> as a sum of contributions from states
PHYSICS LETTERS
Volume 127B, number 6 ]p,) that correspond
to the particle momenta p, (8):
c
(pJBlpa). (11) PC? Each of these contributions are proportional to the quantum-mechanical probability of finding a particle in a state with momentum p,. We note that in any reference frame the basis set of states over which all the necessary averaging is done involves only I&states. The Jp,)-states play a minor role: they enable an expression of the quantities of interest in momentum variables by reexpanding the matrix elements according to (11). hIBId
=
3. To construct the density matrix p we study the properties of the energy and momentum operators of the system (3), (4). We shall assume our box to be large enough so that we can neglect the Casimir energy. Then the energy and the momentum of the field defined over a hypersurface t = const . in an arbitrary reference frame is PQ = j-To0 d3x V
= pi =
Ca;aa [uOea +(k,*u)2/e,uo]tF,O(t, a
...).
.[Toid3, V
The field in a box is not a closed system: Toi # 0 at the boundary. The time-dependent operators F,” are bilinear in aol, ai. We shall not need the explicit form of these operators, so we give only their properties u*F, = 0,
Wqylr) = 0,
(13)
where I$ is an arbitrary N,,,-particle state. The density matrix of a statistical system is well known for a system at rest p =.Z’exp(-H/T),
Spp = 1,
11 August 1983
observer and Z as we have shown in section 1 is invariant. What seems necessary now is to express the Lorentz-invariant operator exp (-H/T) in terms of the quantities concerned with a moving system, H=
c E&z& CI
= c uvPV = ’ U’Pklm N;,m v kJ,m
(k,Z,m=O,+l,...
).
(15)
We can easily check that the latter equality is valid (Pv is not a four-vector!) by making use of the relations (12), (13) and (9). Consider a relativistic partition function ZL of one particle. According to (14), (15) and (9)&(11) we have Z, = c
a!
=
(al exp (-H/T)Id
c 1 p’zm exp[7(u*pklm)/T]. k,l,m=O,~l,... 8 uOu.pklrn (16)
We have thus represented the partition function of one particle as a sum of the contributions from states that correspond to the allowed values of the energy and the momentum of a particle, p$, , (8) in an arbitrary reference frame. According to (lo), (11) each of these contributions is proportional to the probability of finding a particle in a state with momentum pklrn _The representation (16) for a partition function thus guarantees its direct relationship to the particle distribution function in momentum space. Assuming the dimensions of a box to be large enough, we change from summation to integration according (8)
c
k,l,m=O,?l,...
+JJJd3p _-m
The partition function of N particles is then
v,
!?!!.
(W3
PO
(17)
of an ideal Boltzmann
gas
(14)
where H is the hamiltonian. T is the temperature determined in a rest system. We henceforward regard the temperature T as invariant [7]. The probability of a state that has an energy Ei in a rest system cannot depend upon the character of the motion of an
v=
v*/u?
The representation
(18) (18) for a partition
function
is 445
much different from the conventional expression (1) proposed by Touschek [2] . The difference is that the integrand in the partition function (18) in any reference frame coincides, with an accuracy up to an invariant multiplier, with the density of the distribution function F$(...pj . ..) whereas the representation (1) has this property in a rest system only. The relativistic partition function of a Boltzmann gas (18) thus contains all the information needed to calculate thermodynamical quantities in an arbitrary reference frame. The important feature of the representations (16), (18) is to be noted. The point is that every term of the sum (16), taken separately is not a Lorentz-invariant quantity, This agrees with the fact that the oneparticle phase volume element dI’I = Vd3p in (18) is not invariant. Still, the partition functions (16), (18) are invariant. This can easily be seen directly. We now consider the density matrix for a Bosegas. Expanding the diagonal matrix element of the density matrix (14), (15) with the use of (11) we obtain the distribution function w,(...I~~~...) in the occupation number representation nP of states with (Y momenta pal wu(...np
x
*..) =z-l
els of the energy-momentum of a particle in a box, every separate factor is not Lorentz-invariant. Still, we can easily see by transforming the 2 to a product over levels that the partition function (20) is invariant. The procedure of studying the relativistic partition function and the distribution function for a photon gas is basically the same as that given above for scalar particles. Some technical modifications are due to the presence of a spin. It results, finally, in taking trivial account of the level degeneracy multiplicity. The relativistic partition function for a photon gas is expressed through the use of (20) by substituting IIp + IIi=l,211p where the index] = 1,2 corresponds to mutually orthogonal polarizations of a photon. 4. The direct connection of the partition functions (18), (20) with the corresponding distribution functions makes it possible to develop a relativistic method of obtaining thermodynamical and statistical quantities. For this purpose we introduce a generating partition function ZQ,,), where /3p is an arbitrary spacelike four-vector. For a Boltzmann gas we have
sd3Npexp(-8,
$$‘).
(21)
n (c npa)! i
For a Bose gas, when taking into account (17), (20), we have
r$pa!)-(a&p
ln Z”@,>
01
Xexp
-PZ c 8uOT P,
(
“P, )I]a.
(19)
z = {Cl ZwJ...n % = n P
(1 _ exp[-
...) PCY T-1(~.p)])[email protected]).
s
=__!&_
d3p ln[ 1 - z exp (-/3-p)]
(27G3
The partition function of the grand canonical ensemble is obtained by summing the distribution wv(...npQ... ) over all occupation numbers. We now use the multinomial theorem and the relation (lo), obtaining
(20)
The product in (20) is over all allowed values p = pklm (8) in the reference frame under consideration. In the partition function (20) factorized over individual lev446
11 August 1983
PHYSICS LETTERS
Volume 127B, number 6
-
ln[l -z exp(-P-P~~))I.
(22)
We extract a term that corresponds to the lowest level from under the integral sign. In our paper z = 1 ,g is the level degeneracy multiplicity. The generating functions ZU are not Lorentzinvariant for an arbitrary four-vector @‘J.At 0’1 = up/T, however, they are the same as the truly partition functions (18), (20): Z”@, = up/T) = Z(T, . ..) = inv.
(23)
The average values of energy and momentum in some reference frame are expressed in terms of quantities referred to this system
Volume
127B, number
6
PHYSICS
(24) This quantity no longer forms a four-vector. The expression (24) coincides with the results (2) of the relativistic mechanics of continuous media. In the present paper these results are first derived within the framework of relativistic statistical physics, The properties of relativistical statistical and thermodynamical quantities will be studied within the present approach in a separate paper. We here note only the following result. It is easy to see that the entropy S is invariant: Su = -Spplnp =lnZ+
(p C--+v
w =S,=inv.
(25)
It follows from our analysis of the distribution function, however, that the set of states that correspond to maximum entropy in an ideal gas in equilibrium are not related in different reference frames by Lorentztransformations for totally free particles. Since the pressure P is also invariant, a special role in relativistic statistical physics must be played by ensembles with constant pressure. The method developed by us makes it possible to write the generating partition function for an isobaric ensemble (?‘, N, T are fixed):
LETTERS
11 August
statistical and thermodynamical tems with constant pressure.
description
1983
of sys-
5. We conclude by noting the following. The gas in thermodynamical equilibrium cannot be regarded as a closed system in a relativistic situation. The mechanical equilibrium in which the gas happens to be is caused by its interaction with surrounding bodies. This fact may be taken into account effectively by means of corresponding boundary conditions. The present method, enables a relativistic description of a statistical system in an arbitrary inertial system. As distinct from the ideology of ref. [9] all observers are treated on an equal footing. The description is given in terms of quantities measured simultaneously in each system. When we study the properties of hadronic matter the methods of equilibrium statistical physics may be applied for (quasi) equilibrium systems only. This system can, therefore, be only inside a hadron bunch (not at the boundary with vacuum). We thus again come to a relativistically open system, for example, a system under constant pressure, as is proposed in ref. [lo] . The appropriate relativistic formalism is developed in this paper. The present results may serve as a starting point for different generalizations and approximations in the case of systems involving interaction. The author extends his gratitude to M.I. Gorenstein, A.N. Makhlin, V.P. Shelest and G.M. Zinoviev for valuable comments.
-4” = %‘/T,
(26)
where ZI; is defined according to (2 l), A” is the constant required by the relations of dimensionality [8] and invariance of Z;p = Z$(fip = up/T). The transformational properties of the energy and momentum of a gas are defined by (2) with substitution T/ -+ (v). However, the quantity HP = -(a/apJlnZ$l,
zu
M
,*
P
= ((PO) +3 W), W),
(27)
unlike (24), is already a four-vector. This enables an introduction of the invariant enthalpy [.5], H = (H,P)1/2. The entropy S is then S=lnZy+H/T=inv. These relations are initial relations for a relativistic
(28)
[l] R. Hagedorn
[2] [3] [4] [S] [6] [7] [8] [9] [lo]
and J. Rafelski, Phys. Lett. 97B (1980) 136; J. Rafelski and R. Hagedorn, From hadron gas to quark matter, in: Statistical mechanics of quarks and hadrons, ed. H. Satz (North-Holland, Amsterdam, 1981). B. Touschek, Nuovo Cimento 58 (1968) 295. D. Ter Haar and H. Wergeland, Phys. Rep. 1 (1971) 31. M. Planck, Berl. Ber. (1907) p. 542. H. Callen and G. Horwitz, Am. J. Phys. 39 (1971) 938. W. Pauli, Relativitatstheorie, reprint (Moscow, 1947). P.T. Landsberg, Nature (London) 212 (1966) 571; N.G. Van Kampen, Phys. Rev. 173 (1968) 295. J.E. Mayer and M.G. Mayer, Statistical mechanics, 2nd Ed. (Wiley Interscience, New York, 1977). G. Cavalleri and G. Salgarelli, Nuovo Cimento 62A (1969) 722. M.I. Gorenstein, Yad. Fiz. 31 (1980) 1630; R. Hagedorn, On a possible phase transition between hadron matter and quark-gluon matter, preprint TH 3207CERN (1981).
447
127B, number
PHYSICS
6
ON THE RELATIVISTIC
PARTITION
LETTERS
FUNCTION
11 August
1983
OF IDEAL GASES
Yu .M, SINYUKOV USSR State Standards Committee, Moscow, USSR Received
3 December
1982
The covariant partition function method for ideal Boltzmann and Bose gases is developed within quantum field theory. This method is a basis to describe the statistical and thermodynamical properties of the gases in canonical, grand canonical and pressure ensembles in an arbitrary inertial system. It is shown that when statistical systems are described relativistically it is very important to take into account the boundary conditions. This is due to the fact that an equilibrium system is not closed mechanically. The results may find application in hadron physics.
I. The requirement that the descriptions of a thermodynamic system in different inertial systems should correspond to one another unambiguously was long considered to be of fundamental rather than practical character. The problems in hadron physics, however, have now confronted us with the practical need of a consistent relativistic formulation of statistical theory. As an example, we can indicate the last series of papers by Hagedorn and collaborators [ 1 ] where the hadron-quark phase transition was investigated largely based on a covariant formulation of statistical mechanics [2 ] . The reason for writing this paper is that the relativistic formulation of statistical mechanics proposed by Touschek [2] and appearing to be generally accepted (see ref. [3] ) cannot be regarded as physically correct. What we mean is as the following. The Boltzmann equation S = k In r and the invariance of entropy S [4] result in the requirement that the phase volume r‘ = J dF is invariant. To guarantee this, Touschek proposed [2] that the element of phase volume, dI’, should also be invariant. Then a relativistic expression for the phase volume occupied by a gas of N particles with mass m in a canonical ensemble (referred to as the partition function 2) is
where up is the four-dimensional velocity of a thermodynamic system, that moves as a whole (with velocity u); the asterisk denotes the quantities referring to a system at rest. The assumption that the dr is invariant looks natural and is the only means of securing invariance of JdI’. This means, in fact, that all the states of the ideal gas transform according to the usual Lorentz transformation for a totality of free particles. And precisely this assumption that the element dr of a phase volume occupied by an ideal gas is invariant, is incorrect. An ideal gas in a state of global equilibrium should be considered in a container (“box”) of volume V. The phase volume element dr of a system is not invariant because the particles in a box are not free, but influenced by the walls of the container where they happen to be. A nontrivial action of the walls confining the system was clearly manifested in relativistic thermodynamics by Callen and Horwirz in ref. [S] A thermodynamical system in a box is not closed; therefore its energy and momentum no longer form a four-vector. The energy and momentum transformation law for these systems are [6] (9 is the pressure): E = Un(E* + &V*),
exp[-(u’Pi)/T]6(pi2-m2)d4pi, (1) 0 031-9163/83/0000-0000/$03.00
0 1983 North-Holland
P = U(E, + TV*).
(2)
The formulas (2) cannot be explained within the covariant formulation of statistical mechanics [2] , be443
Volume
127B, number
6
PHYSICS
cause in the framework of that approach an ideal equilibrium gas in volume Vis incorrectly regarded as a closed system. We say that the representation (1) for the partition function is not related to a description of a statistical system by a moving observer: the integrand does not correspond to the momentum distribution function in a moving reference frame. 2. Consider a system of free uncharged scalar particles of mass m. in a rectangular box of dimensions a X b X c. The equation for a field and the condition for confining the field in a box are :
(0 -
m&w =0,
q(x) = 0
(3)
at the boundary
and outside the box.
(4)
We note that unlike the condition of “impenetrability” of the walls (4), imposing periodic boundary conditions on the field is not a Lorentzcovariant operation. To construct the basis of the functions we make use of the Lorentz-invariant scalar product
(5) Expanding the arbitrary solution of eqs. (3), (4) on the corresponding complete orthonormalized set of solutions, we have after quantization
Here a:, sol are the annihilation and creation operators of states with quantum numbers (Y. Our task is to study the energy and momentum distribution functions of particles in an arbitrary reference frame. For this purpose we expand the basis functions q(y in plane waves normalized to one particle in a box:
LETTERS
11 August
4&n=u eklm 0
1983
+ kklm- u,
Pklm = UEklm + UOkklm’
k klm = (n-k/a, d/b, nmlc),
eklm = (rnt + k&,)‘/2, (8)
The eight operators aP, are related to a, by “p, = aklm =
i2~312(klm/Iklm~)[p~~,/uO(~~pk~m)] l12aKLM
(aKLM = aor) .
(9
The commutation relations for these operators follow from (6). These commutators are not the standard Bose-commutators because the set {+, } are not the basis functions for eqs. (3), (4). Howe&, the introduction of the operators up is a useful trick. It enables one to express the stagstical quantities in energy-momentum variables. We note in the first place that the expansion (7) is the expansion of wavefunction of the system on the sum of orthonormalized states corresponding at Y113 5 A/lp, 1to particles with momentum poi. The coefficients multiplying aKM in (9) are the coefficients of the expansion of q, on pP . The square of the absolute value of these coeffic&ts is the probability W@,) of finding a particle with momentum p, if the system is in a state (Y.The operator
that acts on states . . . (cz~)~ .. . 10)thus corresponds to the average number of particles with momentum par. From (9) follows the explicit dependence of the operator Nia on the velocity of a system. Using (S), (9), we find that the normalization of the state Ip,) E aiq 10)is the same as the probability of observing a particle in this state (p, Ip,) = WN;ala) = 1
p:
= w&J>
8uowP,)
pppp,=
(2~:abc/uo)-1i2
exp (-ip;x).
(7)
For every level (Y= {K,L,M}, (K,L,M = 0,l . ..) the momentum p, z (pklrn) takes on 8 values correspondingtok=+K,l=+L,m=+M,
444
c W(p,) = 1 PQ!
(a fixed).
(10)
This allows us to write down the contribution to the mean value of some linear operator B from a oneparticle state la> as a sum of contributions from states
PHYSICS LETTERS
Volume 127B, number 6 ]p,) that correspond
to the particle momenta p, (8):
c
(pJBlpa). (11) PC? Each of these contributions are proportional to the quantum-mechanical probability of finding a particle in a state with momentum p,. We note that in any reference frame the basis set of states over which all the necessary averaging is done involves only I&states. The Jp,)-states play a minor role: they enable an expression of the quantities of interest in momentum variables by reexpanding the matrix elements according to (11). hIBId
=
3. To construct the density matrix p we study the properties of the energy and momentum operators of the system (3), (4). We shall assume our box to be large enough so that we can neglect the Casimir energy. Then the energy and the momentum of the field defined over a hypersurface t = const . in an arbitrary reference frame is PQ = j-To0 d3x V
= pi =
Ca;aa [uOea +(k,*u)2/e,uo]tF,O(t, a
...).
.[Toid3, V
The field in a box is not a closed system: Toi # 0 at the boundary. The time-dependent operators F,” are bilinear in aol, ai. We shall not need the explicit form of these operators, so we give only their properties u*F, = 0,
Wqylr) = 0,
(13)
where I$ is an arbitrary N,,,-particle state. The density matrix of a statistical system is well known for a system at rest p =.Z’exp(-H/T),
Spp = 1,
11 August 1983
observer and Z as we have shown in section 1 is invariant. What seems necessary now is to express the Lorentz-invariant operator exp (-H/T) in terms of the quantities concerned with a moving system, H=
c E&z& CI
= c uvPV = ’ U’Pklm N;,m v kJ,m
(k,Z,m=O,+l,...
).
(15)
We can easily check that the latter equality is valid (Pv is not a four-vector!) by making use of the relations (12), (13) and (9). Consider a relativistic partition function ZL of one particle. According to (14), (15) and (9)&(11) we have Z, = c
a!
=
(al exp (-H/T)Id
c 1 p’zm exp[7(u*pklm)/T]. k,l,m=O,~l,... 8 uOu.pklrn (16)
We have thus represented the partition function of one particle as a sum of the contributions from states that correspond to the allowed values of the energy and the momentum of a particle, p$, , (8) in an arbitrary reference frame. According to (lo), (11) each of these contributions is proportional to the probability of finding a particle in a state with momentum pklrn _The representation (16) for a partition function thus guarantees its direct relationship to the particle distribution function in momentum space. Assuming the dimensions of a box to be large enough, we change from summation to integration according (8)
c
k,l,m=O,?l,...
+JJJd3p _-m
The partition function of N particles is then
v,
!?!!.
(W3
PO
(17)
of an ideal Boltzmann
gas
(14)
where H is the hamiltonian. T is the temperature determined in a rest system. We henceforward regard the temperature T as invariant [7]. The probability of a state that has an energy Ei in a rest system cannot depend upon the character of the motion of an
v=
v*/u?
The representation
(18) (18) for a partition
function
is 445
much different from the conventional expression (1) proposed by Touschek [2] . The difference is that the integrand in the partition function (18) in any reference frame coincides, with an accuracy up to an invariant multiplier, with the density of the distribution function F$(...pj . ..) whereas the representation (1) has this property in a rest system only. The relativistic partition function of a Boltzmann gas (18) thus contains all the information needed to calculate thermodynamical quantities in an arbitrary reference frame. The important feature of the representations (16), (18) is to be noted. The point is that every term of the sum (16), taken separately is not a Lorentz-invariant quantity, This agrees with the fact that the oneparticle phase volume element dI’I = Vd3p in (18) is not invariant. Still, the partition functions (16), (18) are invariant. This can easily be seen directly. We now consider the density matrix for a Bosegas. Expanding the diagonal matrix element of the density matrix (14), (15) with the use of (11) we obtain the distribution function w,(...I~~~...) in the occupation number representation nP of states with (Y momenta pal wu(...np
x
*..) =z-l
els of the energy-momentum of a particle in a box, every separate factor is not Lorentz-invariant. Still, we can easily see by transforming the 2 to a product over levels that the partition function (20) is invariant. The procedure of studying the relativistic partition function and the distribution function for a photon gas is basically the same as that given above for scalar particles. Some technical modifications are due to the presence of a spin. It results, finally, in taking trivial account of the level degeneracy multiplicity. The relativistic partition function for a photon gas is expressed through the use of (20) by substituting IIp + IIi=l,211p where the index] = 1,2 corresponds to mutually orthogonal polarizations of a photon. 4. The direct connection of the partition functions (18), (20) with the corresponding distribution functions makes it possible to develop a relativistic method of obtaining thermodynamical and statistical quantities. For this purpose we introduce a generating partition function ZQ,,), where /3p is an arbitrary spacelike four-vector. For a Boltzmann gas we have
sd3Npexp(-8,
$$‘).
(21)
n (c npa)! i
For a Bose gas, when taking into account (17), (20), we have
r$pa!)-(a&p
ln Z”@,>
01
Xexp
-PZ c 8uOT P,
(
“P, )I]a.
(19)
z = {Cl ZwJ...n % = n P
(1 _ exp[-
...) PCY T-1(~.p)])[email protected]).
s
=__!&_
d3p ln[ 1 - z exp (-/3-p)]
(27G3
The partition function of the grand canonical ensemble is obtained by summing the distribution wv(...npQ... ) over all occupation numbers. We now use the multinomial theorem and the relation (lo), obtaining
(20)
The product in (20) is over all allowed values p = pklm (8) in the reference frame under consideration. In the partition function (20) factorized over individual lev446
11 August 1983
PHYSICS LETTERS
Volume 127B, number 6
-
ln[l -z exp(-P-P~~))I.
(22)
We extract a term that corresponds to the lowest level from under the integral sign. In our paper z = 1 ,g is the level degeneracy multiplicity. The generating functions ZU are not Lorentzinvariant for an arbitrary four-vector @‘J.At 0’1 = up/T, however, they are the same as the truly partition functions (18), (20): Z”@, = up/T) = Z(T, . ..) = inv.
(23)
The average values of energy and momentum in some reference frame are expressed in terms of quantities referred to this system
Volume
127B, number
6
PHYSICS
(24) This quantity no longer forms a four-vector. The expression (24) coincides with the results (2) of the relativistic mechanics of continuous media. In the present paper these results are first derived within the framework of relativistic statistical physics, The properties of relativistical statistical and thermodynamical quantities will be studied within the present approach in a separate paper. We here note only the following result. It is easy to see that the entropy S is invariant: Su = -Spplnp =lnZ+
(p C--+v
w =S,=inv.
(25)
It follows from our analysis of the distribution function, however, that the set of states that correspond to maximum entropy in an ideal gas in equilibrium are not related in different reference frames by Lorentztransformations for totally free particles. Since the pressure P is also invariant, a special role in relativistic statistical physics must be played by ensembles with constant pressure. The method developed by us makes it possible to write the generating partition function for an isobaric ensemble (?‘, N, T are fixed):
LETTERS
11 August
statistical and thermodynamical tems with constant pressure.
description
1983
of sys-
5. We conclude by noting the following. The gas in thermodynamical equilibrium cannot be regarded as a closed system in a relativistic situation. The mechanical equilibrium in which the gas happens to be is caused by its interaction with surrounding bodies. This fact may be taken into account effectively by means of corresponding boundary conditions. The present method, enables a relativistic description of a statistical system in an arbitrary inertial system. As distinct from the ideology of ref. [9] all observers are treated on an equal footing. The description is given in terms of quantities measured simultaneously in each system. When we study the properties of hadronic matter the methods of equilibrium statistical physics may be applied for (quasi) equilibrium systems only. This system can, therefore, be only inside a hadron bunch (not at the boundary with vacuum). We thus again come to a relativistically open system, for example, a system under constant pressure, as is proposed in ref. [lo] . The appropriate relativistic formalism is developed in this paper. The present results may serve as a starting point for different generalizations and approximations in the case of systems involving interaction. The author extends his gratitude to M.I. Gorenstein, A.N. Makhlin, V.P. Shelest and G.M. Zinoviev for valuable comments.
-4” = %‘/T,
(26)
where ZI; is defined according to (2 l), A” is the constant required by the relations of dimensionality [8] and invariance of Z;p = Z$(fip = up/T). The transformational properties of the energy and momentum of a gas are defined by (2) with substitution T/ -+ (v). However, the quantity HP = -(a/apJlnZ$l,
zu
M
,*
P
= ((PO) +3 W), W),
(27)
unlike (24), is already a four-vector. This enables an introduction of the invariant enthalpy [.5], H = (H,P)1/2. The entropy S is then S=lnZy+H/T=inv. These relations are initial relations for a relativistic
(28)
[l] R. Hagedorn
[2] [3] [4] [S] [6] [7] [8] [9] [lo]
and J. Rafelski, Phys. Lett. 97B (1980) 136; J. Rafelski and R. Hagedorn, From hadron gas to quark matter, in: Statistical mechanics of quarks and hadrons, ed. H. Satz (North-Holland, Amsterdam, 1981). B. Touschek, Nuovo Cimento 58 (1968) 295. D. Ter Haar and H. Wergeland, Phys. Rep. 1 (1971) 31. M. Planck, Berl. Ber. (1907) p. 542. H. Callen and G. Horwitz, Am. J. Phys. 39 (1971) 938. W. Pauli, Relativitatstheorie, reprint (Moscow, 1947). P.T. Landsberg, Nature (London) 212 (1966) 571; N.G. Van Kampen, Phys. Rev. 173 (1968) 295. J.E. Mayer and M.G. Mayer, Statistical mechanics, 2nd Ed. (Wiley Interscience, New York, 1977). G. Cavalleri and G. Salgarelli, Nuovo Cimento 62A (1969) 722. M.I. Gorenstein, Yad. Fiz. 31 (1980) 1630; R. Hagedorn, On a possible phase transition between hadron matter and quark-gluon matter, preprint TH 3207CERN (1981).
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