The Time Arrow in Planck Gas

Miroslaw Kozlowskia, Janina Marciak-Kozlowskab, c

a  Institute of Experimental Physics, Warsaw University, Ho\.za 69, 00-681 Warsaw, Poland

b Institute of Electron Technology, Al. Lotników 32/46, 02-668 Warsaw, Poland

 

c Author to whom correspondence should be addressed.

Abstract

In this paper the quantum heat transport equation (QHT) is applied to the study of thermal properties of Planck gas, i.e.  gas of the massive particles with mass equal Planck mass, Mp=((h/2p)c/G)1/2 and relaxation time equals Planck time, tp=((h/2p)G/c5)1/2. The quantum of thermal energy for Planck gas, EPlanck=1019 GeV and quantum thermal diffusion coefficient DPlanck=((h/2p) G/c)1/2 are calculated. Within the framework of QHT the thermal phenomena in Planck gas can be divided into two classes, for time period shorter than tp the time reversal symmetry holds and for time period longer than tp, time symmetry is broken, i.e. time arrow is created.

Key words:   Planck gas; Quantum heat transport; Time arrow

1  Introduction

The enigma of Planck Era, i.e. the event characterized by Planck time, Planck radius and Planck mass is a very attractive for speculations. In this paper we discuss the new (as we think) interpretation of the Planck time. We define Planck gas - the gas of massive particles all with masses equal Planck mass Mp=((h/2p) c/G)1/2 and relaxation for transport process equals the Planck time: tp=((h/2p) G/c5)1/2. To the description of the thermal transport process in Planck gas we apply the quantum heat transport equation (QHT) derived in our earlier paper [1]. The QHT is the specification of the hyperbolic heat conduction equation HHC [1,2] to the quantum limit of heat transport i.e. when de Broglie wave length, lB equals mean free path, l. When QHT was applied to the description of thermal excitation of the matter it was shown that the excited matter response is quantized on the different levels (atomic, nuclear) with quantum thermal energy equal Eatomic ~ 9 eV, Enuclear ~ 7 MeV. In this paper using QHT we calculate quantum thermal energy for Planck gas the heaton , EPlanck ~ 1019 GeV and quantum diffusion coefficient for Planck gas
DPlanck= æ
ç
è 		(h/2p) G
c
 ö
÷
ø 		1/2

 
 .

The QHT for Planck gas is the damped wave equation which for time period Dt ~ tp is the hyperbolic wave equation with preserved time reversal symmetry. On the other hand for time period Dt >> tp the QHT is parabolic diffusion equation with broken time symmetry. It seems that Planck time tp divides the transport phenomena on two classes: for pre -Planck times the time reversal symmetry holds and for post -Planck time the time symmetry is broken, i.e. time arrow is created.

2  Thermal properties of the Planck gas

In the following we will describe the thermal properties of the Planck gas. To that aim we use hyperbolic heat transport equation (HHC) [1]
lB
vh
    2T
t2
 + lB
l
    T
t
 = (h/2p)
Mp
  Ñ2T.
(1)
In equation (1) Mp is the Planck mass lB - de Broglie wave length and l - mean free path for Planck mass. The HHC equation describes the dissipation of the thermal energy induced by temperature gradient ÑT. Recently the dissipation processes in the cosmological context (e.g. viscosity) were described in the frame of EIT (Extended Irreversible Thermodynamics) [2,3]. With the simple choice for viscous pressure it is shown that dissipative signals propagate with the light velocity, c [2]. Considering that the relaxation time t is defined as [1],
t = (h/2p)
Mpvh2
 ,
(2)
for thermal wave velocity vh=c, one obtains
t = (h/2p)
Mpc2
 = æ
ç
è 		(h/2p) G
c5
 ö
÷
ø 		1/2

 
 =tp,
(3)
i.e. the relaxation time is equal the Planck time tp. The gas of massive particles with masses equal Planck mass Mp and relaxation time for transport processes equals Planck time tp we will define as the Planck gas.

According to the result of the paper [1] we define the quantum of the thermal energy, the heaton for the Planck gas
Eh
=
(h/2p)w = (h/2p)
tp
 = æ
ç
è 		(h/2p)c
G
 ö
÷
ø 		1/2

 
 c2=Mpc2,
Eh
=
Mpc2=EPlanck=1019 GeV.
(4)
With formula (2) and vh=c we calculate the mean free path, l, viz.
l = vhtp=ctp=c æ
ç
è 		(h/2p)G
c5
 ö
÷
ø 		1/2

 
 = æ
ç
è 		(h/2p) G
c3
 ö
÷
ø 		1/2

 
 .
(5)
From formula (5) we conclude that mean free path for Planck gas is equal Planck radius. For Planck mass we can calculate the de Broglie wave length
lB= (h/2p)
Mpvh
 = (h/2p)
Mpc
 = æ
ç
è 		G(h/2p)
c3
 ö
÷
ø 		1/2

 
 =l.
(6)
As it is defined in paper [1] equation (6) describes the quantum limit of heat transport. When formulae (5), (6), are substituted to the equation (1) we obtain:
tp  2 T
t2
 + T
t
 = (h/2p)
Mp
  Ñ2T.
(7)
Equation (7) is the quantum hyperbolic heat transport equation (QHT) for Planck gas. Equation (7) can be written as:
2T
t2
 + æ
ç
è 		c5
(h/2p)G
 ö
÷
ø 		1/2

 
 T
t
 =c2Ñ2T.
(8)
It is interesting to observe that QHT is the damped wave equation and gravitation influences the dissipation of the thermal energy. In paper [4] P. G. Bergmann discussed the conditions for the thermal equilibrium in the presence of the gravitation. As it was shown in that paper, the thermal equilibrium of spatially extended systems is characterized by the ``global'' temperature and a ``local'' temperature which is sensitive to the value of the gravitational potential.

On the other hand equation (8) describes the correlated random walk of Planck mass. For mean square displacement of random walkers we have
áx2ñ = 2(h/2p)
Mp
 é
ê
ë 		t
tp
 -(1-e-t/tp) ù
ú
û 		.
(9)
From formula (6), we conclude that, for t ~ tp,
áx2ñ @ (h/2p)
Mpt
  t2
(10)
or
áx2ñ @ c2t2,
(11)
and we have thermal wave with velocity c. For t >> tp, we have
áx2 ñ ~ 2(h/2p)t
Mp
   æ
ç
è 		t
tp
 -1 ö
÷
ø 		= 2(h/2p)
Mp
 t=2DPlanckt,
(12)
where
DPlanck= (h/2p)
Mp
 = æ
ç
è 		(h/2p)G
c
 ö
÷
ø 		1/2

 
(13)
denotes the diffusion coefficient for Planck mass.

We can say that for time period of the order of Planck time QHT describes the propagation of thermal wave with velocity equal c and for time period much longer than tp QHT describes the diffusion process with diffusion coefficient dependent on gravitation constant G.

The quantum hyperbolic heat equation (7) as a hyperbolic equation shed a light on the time arrow in Planck gas. When QHT is written in the equivalent form
tp  2T
t2
 + T
t
 = æ
ç
è 		(h/2p) G
c
 ö
÷
ø 		1/2

 
 Ñ2T,
(14)
then, for time period shorter than tp, we have preserved time reversal for thermal processes, viz,
1
c2
    2T
t2
 =Ñ2T,
(15)
and, for t >> tp,
T
t
 = æ
ç
è 		(h/2p)G
c
 ö
÷
ø 		1/2

 
 Ñ2T,
(16)
the time reversal symmetry is broken.

These new properties of QHT open new possibilities for the interpretation of Planck time. Before tp thermal processes in Planck gas are symmetrical in time. After tp the time symmetry is broken. Moreover it seems that gravitation is activated after tp and this fact creates time arrow (formula (16)).

3  Conclusions

In this paper the thermal properties of the Planck gas are discussed. We have developed quantum heat transport equation (QHT) for Planck gas, with quantum thermal diffusion coefficient DPlanck=((h/2p) G/c)1/2. The quantum of the thermal energy, the heaton for Planck gas is defined and calculated, EPlanck=1019 GeV. It is shown that for t > Planck time, the time symmetry is broken.

Acknowledgement   I thank the referee, unknown to me, who brought reference [3], to my attention as well suggesting alternative explanation for formula (3). This study was made possible by financial support from Polish Committee for Science Research under grant No 8T11B 046 09 and 8T11B 002 12.

References

[1]
Marciak-Kozlowska, J., Kozlowski, M., Foundations of Physics Letters 9, 235 (1996)
[2]
Jou, D., Casas-Vasquez, J., Lebon, G., Extended Irreversible Thermodynamics (Springer, Berlin, 1993)
[3]
Maartens, R., Class. Quantum Grav. 12, 1455 (1995)
[4]
Bergmann, P. G. in Cosmology and Particle Physics V. de Sabbata and H. Tso-Hsiu (eds) Kluwer Academic Publishers, 1994, p. 9


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