Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility
The introduction of a discrete space-time represents an attempt to describe the physics at the Planck’s scale. We show that this concept can be also very useful in describing thermodynamics in a pre-relativistic world. From this concept a new approach of statistical mechanics based on a dynamic...
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Інститут фізики конденсованих систем НАН України
2003
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| Цитувати: | Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility / J.P. Badiali // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 375-386. — Бібліогр.: 19 назв. — англ. |
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irk-123456789-1207562017-06-13T03:02:50Z Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility Badiali, J.P. The introduction of a discrete space-time represents an attempt to describe the physics at the Planck’s scale. We show that this concept can be also very useful in describing thermodynamics in a pre-relativistic world. From this concept a new approach of statistical mechanics based on a dynamic viewpoint and an entropy representation is presented. The entropy is connected with the counting of the paths in space-time. It contains a time interval that represents the time that we have to wait in order to relax the quantum fluctuations and to reach the thermal regime. It is shown that this time is β~ . The mathematical expressions we derive for thermal quantities like the entropy and the free energy are identical to those obtained by the traditional path-integral formalism starting from the canonical form of the thermal density matrix. However, the introduction of a quantized spacetime shows that thermodynamics is consistent with an equation of motion that is time-irreversible at a microscopic level. As a consequence, the problem of irreversibility is revisited and the derivation of a H-theorem becomes possible in the future Введення дискретного простору-часу є спробою опису фізики в масштабі Планка. Нами буде показано, що така концепція може також бути корисною для опису термодинаміки в пре-релятивістичній системі. Виходячи з цієї концепції, нами запропоновано новий статистико-механічний підхід, що базується на динамічній точці зору і представленні ентропії. Ентропія пов’язана з процедурою розрахунку шляхів в просторі-часі. Вона містить часовий інтервал, що являє собою час необхідний для загасання квантових флуктуацій і досягнення термічного режиму. Показано, що β~ є цим часом. Математичні вирази, отримані нами для таких термічних величин як ентропія і вільна енергія, є ідентичними з такими, отриманими з традиційного формалізму інтегрування по траекторіях, починаючи з канонічної форми термічної матриці густини. Проте, введення квантованого простору-часу показує, що термодинаміка узгоджується з рівнянням руху, яке є незворотнім в часі на мікроскопічному рівні. Як наслідок, знову постає проблема необоротності і вивід Н-теореми стає можливим у майбутньому. 2003 Article Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility / J.P. Badiali // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 375-386. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 03.65.Ca, 05.30.-d, 05.70.-a, 47.53.+n DOI:10.5488/CMP.6.3.375 http://dspace.nbuv.gov.ua/handle/123456789/120756 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The introduction of a discrete space-time represents an attempt to describe
the physics at the Planck’s scale. We show that this concept can be also
very useful in describing thermodynamics in a pre-relativistic world. From
this concept a new approach of statistical mechanics based on a dynamic
viewpoint and an entropy representation is presented. The entropy is
connected with the counting of the paths in space-time. It contains a time
interval that represents the time that we have to wait in order to relax the
quantum fluctuations and to reach the thermal regime. It is shown that this
time is β~ . The mathematical expressions we derive for thermal quantities
like the entropy and the free energy are identical to those obtained by the
traditional path-integral formalism starting from the canonical form of the
thermal density matrix. However, the introduction of a quantized spacetime
shows that thermodynamics is consistent with an equation of motion
that is time-irreversible at a microscopic level. As a consequence, the problem
of irreversibility is revisited and the derivation of a H-theorem becomes
possible in the future |
| format |
Article |
| author |
Badiali, J.P. |
| spellingShingle |
Badiali, J.P. Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility Condensed Matter Physics |
| author_facet |
Badiali, J.P. |
| author_sort |
Badiali, J.P. |
| title |
Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility |
| title_short |
Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility |
| title_full |
Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility |
| title_fullStr |
Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility |
| title_full_unstemmed |
Statistical mechanics in a discrete space-time. Thermodynamics and time-irreversibility |
| title_sort |
statistical mechanics in a discrete space-time. thermodynamics and time-irreversibility |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120756 |
| citation_txt |
Statistical mechanics in a discrete
space-time. Thermodynamics and
time-irreversibility / J.P. Badiali // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 375-386. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT badialijp statisticalmechanicsinadiscretespacetimethermodynamicsandtimeirreversibility |
| first_indexed |
2025-07-08T18:31:38Z |
| last_indexed |
2025-07-08T18:31:38Z |
| _version_ |
1837104626897256448 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 3(35), pp. 375–386
Statistical mechanics in a discrete
space-time. Thermodynamics and
time-irreversibility
J.P.Badiali
LECA, ENSCP-Université Pierre et Marie Curie,
4 Place Jussieu, 75230 Paris Cedex 05, France
Received April 29, 2003, in final form July 21, 2003
The introduction of a discrete space-time represents an attempt to describe
the physics at the Planck’s scale. We show that this concept can be also
very useful in describing thermodynamics in a pre-relativistic world. From
this concept a new approach of statistical mechanics based on a dynam-
ic viewpoint and an entropy representation is presented. The entropy is
connected with the counting of the paths in space-time. It contains a time
interval that represents the time that we have to wait in order to relax the
quantum fluctuations and to reach the thermal regime. It is shown that this
time is β~ . The mathematical expressions we derive for thermal quantities
like the entropy and the free energy are identical to those obtained by the
traditional path-integral formalism starting from the canonical form of the
thermal density matrix. However, the introduction of a quantized space-
time shows that thermodynamics is consistent with an equation of motion
that is time-irreversible at a microscopic level. As a consequence, the prob-
lem of irreversibility is revisited and the derivation of a H-theorem becomes
possible in the future.
Key words: statistical mechanics, thermodynamics, time-irreversibility,
discrete space-time
PACS: 03.65.Ca, 05.30.-d, 05.70.-a, 47.53.+n
1. Introduction
It is a great pleasure for me to dedicate this paper to my friend Professor My-
roslav Holovko for his 60th birthday celebration. His contribution to statistical me-
chanics and liquid state theory is important and, in parallel, the recent progresses
obtained by his former students in Lviv as well as in other countries across the world
illustrate the importance of the Ukrainian school in statistical mechanics.
Despite the large successes in statistical physics obtained during the last century
some basic problems remain unsolved. The existence and the demonstration of the
H-theorem is one of them. Another problem concerns our real understanding of the
c© J.P.Badiali 375
J.P.Badiali
black hole entropy; here we have to deal with the consistency between quantum
field theory and general relativity, or the so-called quantum gravitation problem.
This paper is an attempt to progress in the derivation of a H-theorem by using
ingredients developed in the domain of quantum gravitation.
It is well known that the canonical quantum gravitation theory leads to some
infinities that we cannot renormalize. This suggests the introduction of new descrip-
tions for the universe at the Planck’s scale. One possible choice consists in aban-
doning the concept of point particles in quantum field theory and in dealing with
superstring theories [1]. Another possibility is to replace the differentiable manifold
of the general relativity by a discrete space-time structure [2,3]. In this paper we
focus on the second alternative and we show that the existence of a discrete space-
time may be a useful concept to describe the physical world at a scale much larger
than the Planck’s scale. More precisely, we want to show that the thermodynamics
of equilibrium is consistent with statistical mechanics in which there exists a motion
that is time-irreversible at the microscopic level. This leads to the development of a
new viewpoint concerning the problem of the arrow of time in thermodynamics (for
a review in this field see for instance [4]).
In the Clausius version, the second law of thermodynamics asserts the existence
of a state function, i.e., the entropy, that is a non-decreasing function of time for
any closed system [5]. The difficulty in the derivation of this law is the inadequacy
between the postulated time-reversible behavior for the motion of particles at a
microscopic level and the observable irreversible behavior of macroscopic systems.
An illustration of this is the problem of the Poincaré recurrence time associated with
the existence of an Hamiltonian [4]. In what follows the motion we consider does
not result from a given Hamiltonian but is the consequence of discreteness of the
space-time. It is due to this that the problem of the arrow of time is revisited.
More than thirty years ago Feynman presented some doubts concerning the con-
tinuum nature of space-time. In a seminal paper Finkelstein [2] has presented the
dilemma of the discrete versus the continuous space-time structure, a problem al-
ready considered by Riemann in the classical world. Finkelstein, after some others,
developed the idea that the topology of space-time is determined by an order struc-
ture rather than by a metric structure. Later, the idea of a discrete space-time has
been extensively developed via the concept of causal sets [3] for which there are
two basic ingredients: the discreteness of space-time and the existence of a causal
structure. In parallel to this causal sets theory, in the domain of string theory, a
minimal length uncertainty relation related to the fact that strings cannot probe
distances below the string scale [6,7] is introduced. Thus, the existence of a minimal
length appears to be a more and more common idea today.
Hereafter, a discrete space-time means that any length is built up from a finite
number of elementary lengths ∆x and that the flow of time occurs in a series of
individual “ticks” of duration ∆t [8]. If we accept the existence of a discrete space-
time we have to decide how it is organized [9]. A quite natural assumption is that
some basic ingredients of the discrete space-time organization are already familiar
to us from our study of the macroscopic world. Hereafter, the problem is that at the
376
Statistical mechanics in a discrete space-time
level of the discrete space-time we have to introduce the ingredients we need from the
quantum theory in order to derive standard thermodynamics. In what follows, we
define the structure of space-time by a relation between space and time that mimics
the uncertainty relations. This seems a quite natural choice because these relations
are the basic ingredient on which the quantum mechanics is built up. As we shall see
the space-time structure induces a given motion and we may define some quantities
as an average over the paths. These averages can be associated with thermodynamic
properties. An entropy associated with the space-time structure is introduced but the
counting of equilibrium states that appears in a standard approach is now replaced
by a counting over the paths in space-time. The tools on which our new approach
of statistical physics is based are the path-integral formalism [10] and the entropy
representation of statistical mechanics [11].
In section 2 we define the quantized space-time and we describe the motion
associated with it. In section 3 we characterize this space-time by counting the
number of closed paths associated with a given time interval τ . For a given value of
the internal energy U we define an entropy on the space-time. In section 4 we derive
the value of τ and we show that our dynamic approach gives the same mathematical
expressions as the ones obtained by Feynman and Hibbs [10]. However, our results
have a different physical meaning and they show the consistency between equilibrium
thermodynamics and a motion that is time-irreversible at the microscopic level. This
result is also discussed in section 4. In the last section we give a short conclusion
and present some new perspectives.
2. Introduction of a discrete and quantized space-time
We first consider the case of a point particle of mass m. We assume that m
induces a discrete structure in space and time. As discussed in the Introduction, this
means that the space-time is formed by a set of points (xi, ti). In this first attempt,
in order to be illustrative we assume that the points are located on the sites of
a regular lattice. Moreover, we assume that the lattice spacings ∆x for the spatial
coordinates and ∆t for the time axis are not independent. At this level of description
the only relation that we can find between ∆x and ∆t is (∆x)2/∆t = ~/m.
Whatever the values of ∆x and ∆t we immediately have
∆x∆p = ~ and ∆t∆E = ~/2 (1)
provided we use ∆p = m(∆x/∆t) and ∆E = (1/2)m(∆x/∆t)2. The relations (1)
mimic the Heisenberg uncertainty relations but they result from the discreteness of
the space-time. Note that these two relations appear here on the same footing as
the consequence of (∆x)2/∆t = ~/m. In standard quantum mechanics the position-
momentum and the time-energy uncertainty relations are not of the same nature
since the second one is not connected with the non-commutation of two operators
[12].
By definition, a path corresponds to a set of sites (xi, ti). We only retain the
paths for which the values of ti form a sequence of ordered numbers in such a way
377
J.P.Badiali
that ti+1 > ti whatever i. There is no such a constraint for the coordinate positions
xi but we consider that xi+1 is necessarily one of the nearest neighbours of xi. Thus,
to any path there is associated a random walk in the usual sense [13]. In principle,
in order to describe the properties of paths we have to work with finite topological
spaces as developed in [9] and we also have to find an alike correspondence principle
showing in which limit the discrete space-time may appear as a continuum manifold
as shown in [14], for instance. Fortunately, our problem is more simple here since
we do not focus on a truly relativistic system.
The relation (∆x)2/∆t = ~/m is essential in what follows, but the precise values
of ∆x and ∆t are not fixed by the uncertainty relations (1). In the pre-relativistic do-
main the velocity of light goes to infinity. Accordingly, the Compton wavelength goes
to zero and no particles can be created by quantum fluctuations. We may assume
that both ∆x and ∆t tend to zero provided we keep the relation (∆x)2/∆t = ~/m.
In such conditions the random walk leads to a continuous diffusion process [13] for
which the diffusion coefficient, D = ~/2m, has a pure quantum origin. This ran-
dom walk can be characterized by a continuous function, q(t0, x0; t, x) representing
the density of transition probability to go from the initial space-time point (t0, x0)
to (t, x) provided that t > t0. From q(t0, x0; t, x) and a real-valued function φ0(x)
defined for (t = t0) we may form a real-valued function φ(t, x) according to
φ(t, x) =
∫
φ0(y)q(t0, y; t, x)dy (2)
which is the solution of the diffusion equation
−∂φ(t, x)/∂t + D∆xφ(t, x) = 0 (3)
verifying the initial-value problem φ(t0, x) = φ0(x). In (3) ∆x means the Laplacian
operator taken at the point x.
In the presence of an external potential, u(t, x), we generalize (3) into
−∂φ(t, x)/∂t + D∆φ(t, x) −
1
~
u(t, x)φ(t, x) = 0. (4)
In contrast to the case of free diffusion (3), the fundamental solution of (4) cannot
be normalized in general [15]. Thus, q(t0, x0; t, x) is no more a density of transition
probability but it verifies the Chapman-Kolmogorov law of composition [15]
q(t1, x1; t2, x2) =
∫
dx3q(t1, x1; t3, x3)q(t3, x3; t2, x2) (5)
provided t1 < t3 < t2 and, therefore, it can be used to describe the transitions in
space-time. The evolution in space-time is then represented by a semi-group.
At this point we have to note several important points. The equation (3) has
the classical form of a diffusion equation but the diffusion coefficient D has a pure
quantum origin D → 0 if ~ → 0. Up to now we consider the paths as the trace of a
motion in real time, this motion is induced by the space-time underlying structure
378
Statistical mechanics in a discrete space-time
and it is not connected with a given Hamiltonian. The function φ(t, x) is a real-
valued function. Therefore, φ(t, x) is not a wave function and it does not have to
verify additional constraints. For example, it does not have to be square integrable.
Finally φ(t, x) is defined for t > t0 and the equation (3) is time-irreversible.
By using the Feynman-Kac formula, the fundamental solution of (4) can be
written in terms of path integral. Then q(t0, x0; t, x) appears as a weighted sum of
all the paths x(t) connecting the points x0 to x during a time interval (t − t0); the
weight of a path is determined by
A[x(t); t, t0] =
t
∫
t0
[
1
2
m
(
dx(t′)
dt′
)2
+ u(t′, x(t′))
]
dt′, (6)
and we have
q(t0, x0; t, x) =
∫
Dx(t) exp
{
−
1
~
A[x(t); t, t0]
}
. (7)
We may observe that the integrand in (6) looks like the Hamiltonian for a free par-
ticle in an external potential. Therefore, A[x(t); t, t0] can be called the Hamiltonian
action or, in quantum field theory, the Euclidean action. However, A[x(t); t, t0] is a
formal writing [16]. To calculate the path integral we have to discretize A[x(t); t, t0]
and the paths that contribute to the path integral are those for which there is
nowhere a derivative i.e. no velocity in the usual sense [17,18].
3. Entropy in the space-time
On the space-time introduced in the previous section we may define some func-
tions provided we are able to associate to each of them the time on which we have to
observe the motion. Amongst all the possible ways of characterizing the space-time
we would like to consider the one which is as close as possible from the standard
thermodynamics.
For equilibrium states the external potential u(t, x) is not an explicit function of
time and as a consequence A[x(t); t, t0] becomes a function of (t − t0) that we note
A[x(t); t−t0]. Let us consider a given space-time point (t0, x0). A way to characterize
this point is to count the number of closed paths around it; we note A0[x(t); t − t0]
the corresponding Euclidean action. If there is only one closed path around (t0, x0)
we may conclude that there is a strict order around this point. Of course, this is a
limiting case in which all the fluctuations around this path should be forbidden. For
actual systems we may choose a mean path selected from (6) by using, for instance,
the steepest descent method but this path does not realize all the physical content
of (6), we also have to consider the fluctuations around this mean path [10]. The
larger is this number of paths the larger are the acceptable fluctuations around the
mean path and the less defined is the structure around (x0, t0).
In the spirit of thermodynamics we may consider that the system in the presence
of an external field is constrained from its preparation. For instance, we can prepare
the system in such a way that it has a precise value for the internal energy U . In
379
J.P.Badiali
parallel to the Euclidean action A0[x(t); t− t0] we define a thermal action according
to (t − t0)U and we can count the paths for which the Euclidean action does not
deviate too much from the thermal action. This leads to weighting the paths using
[A0[x(t); t − t0] − (t − t0)U ] instead of A0[x(t); t − t0]. Let us define the quantity
Γ(t − t0) =
∫
dx0
∫
Dx(t) exp
{
−
1
~
Ao[x(t); (t − t0)] − (t − t0)U ]
}
. (8)
In order to associate Γ(t − t0) with a thermal quantity we assume that there is a
characteristic time interval, τ , associated with thermodynamics and on which we
have to count the paths to be in thermal equilibrium. By analogy with standard
thermodynamics we define an entropy for the paths, Spath, according to
Spath = kB ln Γ(τ) = kB ln
∫
dx0
∫
Dx(t) exp
{
−
1
~
[Ao[x(t); τ ] − τU ]
}
. (9)
This entropy is the starting point on which our statistical mechanics will be based.
The definition of entropy via (9) requires some comments. In (9), the internal
energy, U , is strictly fixed via the mode of preparation and there is no temperature
but a dynamic quantity, τ . Because there is no Gibbs ensemble behind our definition
of entropy we cannot relate (9) to canonical or micro-canonical representation of
statistical mechanics. In fact, (9) is an entropy representation of statistical mechanics
as defined by Callen [11]. Note that the Planck derivation of the black body radiation
law is based on such an entropy representation. However, the difference between (9)
and the Planck derivation is that we count paths in space-time instead of equilibrium
states. The fact that the temperature is not explicit in (9) does not imply that the
system is without temperature, we may define a temperature Tpath according to the
usual thermodynamic relation
dSpath
dU
=
1
Tpath
. (10)
Our definition of the entropy based on dynamic processes in space-time contains τ ,
which has no equivalent in a standard approach. This time interval appears more
explicitly if we rewrite Spath according to
Spath =
kBτ
~
U + kB ln Zpath (11)
with
Zpath =
∫
dx0
∫
Dx(t) exp
{
−
1
~
qAo[x(t); τ ]
}
=
∫
dx0q(0, x0; τ, x0) (12)
in which we have taken t0 = 0. We may interpret Zpath as the total number of closed
paths that we may count during τ , and q(0, x0; τ, x0) is the fundamental solution of
(4) associated with closed paths and a time interval τ . Note that Zpath depends on
τ but not explicitly on U in contrast with Spath, which is a function of U and τ .
380
Statistical mechanics in a discrete space-time
However, from the usual thermodynamics and also from the invariance properties of
the entropy we know that S must depend on U only [19] and the same is expected
for Spath. Therefore, U and τ are not independent.
From (11) and (10) we immediately have
1
Tpath
=
kBτ
~
+
kB
~
(
U + ~
d
dτ
ln Zpath
)
dτ
dU
. (13)
In standard thermodynamics the relation (10) shows that a specific system in which
the total internal energy U is fixed will take a given temperature. Here, to use (10)
we see from (13) that we have to give a meaning to d/dτ ln Zpath. In order to do
that we use the expression of Zpath in terms of q(0, x0; τ, x0) given in (12). From the
Chapman-Kolmogorov law of composition we may write
q(0, x0; τ, x0) =
∫
dxbq(0, x0; δt, xb)q(δt, xb; τ − δt, x0) (14)
which is true for any δt such as 0 < δt < τ . From (12) and (14) we deduce
~
d
dτ
ln Zpath =
~
Zpath
∫
dx0
∫
dxbq(0, xa; δt, xb)
d
dτ
q(δt, xb; τ − δt, x0) (15)
which we can rewrite by using (4)
~
d
dτ
ln Zpath =
~
Zpath
∫
dx0
∫
dxbq(0, x0; δt, xb)[D∆x0
−
1
~
u(xa)]q(δt, xb; τ − δt, x0).
(16)
In this equation the term associated with the external potential can be written
1
Zpath
∫
dx0u(x0)q(0, x0; τ, x0) =
∫
dx0〈uP(x0)〉path (17)
in which we may interpret 〈uP(x0)〉path as the potential energy at the point x0 cal-
culated as an average over the paths. To calculate the contribution of the Laplacian
in (16) we may use the fact that (14) holds if δt → 0. In that case the Laplacian
operates only on the term which looks like the kinetic energy in Ao[x(t); τ ]. A simple
calculation shows that this term is
1
Zpath
∫
dx0
∫
d(δx)q(0, x0; δt, x0 + δx)
[
m
2
(δx
δt
)2
−
~
2δt
]
q(δt, x0 + δx; τ − δt, x0)
(18)
which we can rewrite as
∫
dxa
[
m
2
〈(δx
δt
)2〉
path
−
~
2δt
]
. (19)
This expression represents the difference between the kinetic energy calculated as
an average over the paths and the quantum fluctuation of energy corresponding to a
381
J.P.Badiali
time interval δt on which the kinetic energy is calculated. In the limits δx → 0 and
δt → 0 we may replace q(0, x0; δt, x0 + δx) in (18) by the free particle approximation
which only depends on δt and δx. Moreover, we may approximate q(δt, x0 + δx; τ −
δt, x0) by q(0, x0; τ, x0) and its integral over x0 gives Zpath as shown in (12). In the
limits that we consider it is easy to see that m/2〈(δx/δt)2〉path = ~/2δt. We may also
verify this result by performing an exact calculation but using an explicit form of
u(x). Note that such a result is also expected from (1); when δt → 0 we must recover
the results of the initial lattice on which we have immediately 1/2m(δx/δt)2 = ~/2δt
Thus, the term in brackets in (19) is well defined in the limit δt → 0. More generally,
we write
m
2
〈(δx
δt
)2〉
path
=
~
2δt
− 〈uK(x0)〉path (20)
in which 〈uK(x0)〉path is a well behaved function in the limit δt → 0. In principle
〈uK(x0)〉path can be calculated for a given external potential u(x). For instance, in
the case of quantum oscillators of frequency ω we have
∫
dx0〈uK(x0)〉path =
~ω
4
(
2
exp τω − 1
+ 1
)
. (21)
In that case this integral is identical to the integral of 〈uP(x0)〉path defined in (17).
Finally we can rewrite (13) according to
~
kBTpath
= τ +
[
U −
∫
dx0(〈uK(x0)〉path + 〈uP(x0)〉path)
] dτ
dU
. (22)
This equation is the main result of this section. It establishes a relation between U ,
τ , several quantities calculated over the paths and Tpath.
4. Thermodynamics and characteristic time
Up to now we have defined some quantities like entropy, energies and temperature
that are similar to the ones defined in standard thermodynamics. However, instead
of considering equilibrium states we investigate the properties of closed paths. A
dynamic point of view has been developed in which there exists a characteristic
time, τ , defined by (22). Now we claim that the value of τ is such that we are able
to recover the standard thermodynamics. This means that all the quantities with
the subscript “path” have the same value as in standard thermodynamics although
they have a different physical content.
If Spath corresponds to S we may identify Tpath and T according to (10). We
consider that τ is such that
∫
dxa[〈uK(xa)〉path + 〈uP(xa)〉path] = −~
d
dτ
ln Zpath (23)
which looks like the internal energy is indeed equal to the total internal energy of
the system energy i.e. U . Thus, whatever the value of the derivative dτ/dU from
382
Statistical mechanics in a discrete space-time
(22) we deduce that
τ =
~
kBT
= β~ . (24)
Now, from (23) and (24) we have
~
d
dτ
ln Zpath =
d
dβ
ln Zpath = −U. (25)
This is the usual relation between internal energy U and the partition function and
we have to conclude that Zpath must be identical to Z. Using (24) we may rewrite
the partition function calculated over paths (12) according to
Zpath =
∫
dxa
∫
Dx(t) exp−
1
~
β~
∫
0
[1
2
m
(dx(t′)
dt′
)2
+ u(x(t′))
]
dt′. (26)
We can check that this expression is identical to the one obtained in [10] starting from
the canonical form of the density matrix. If we define F according to F = −kBT ln Z
we see that (11) corresponds to the usual relation F = U − TS. Thus, our results
obtained from an entropy representation are identical to those derived from the
standard canonical formalism. This shows that we have obtained an alternative
description of the thermodynamics directly in terms of motion in space-time. In this
approach the time is a real time and the motion is not due to a given Hamiltonian but
results from the discreteness of the space-time. However, our path integral formalism
represents more than a simple alternative description of thermodynamics as we shall
see below.
In the standard path integral formalism β~ is considered to be a formal time
[10]. In our approach we have first introduced a given time τ and, by combining
dynamics (4) and thermodynamics via (11) and (10), we have shown that τ = β~.
Therefore, we may conclude that τ has a strong physical meaning. A first discussion
regarding this point has been already presented in [18]. Here τ appears to be the
relaxation time that we have to wait in order to relax the quantum fluctuations and
to reach the thermal regime. From standard textbooks in statistical mechanics [5]
it is well known that such a time exists and as a consequence there is no entropy
on a short period of time. However this time plays no role in the Gibbs ensemble
approach for which there is no dynamics.
However, the most important result of our approach is that equilibrium thermody-
namics is consistent with a motion at the microscopic scale which is time-irreversible
since given by (4). This consistency can be explained as follows. Let us consider a
system at equilibrium with a given entropy S and we investigate its fluctuations
starting from an initial time t = 0. By convention, we can say that the entropy at
the initial time is given by S = kB ln Γ where Γ is given by (9). However, to calcu-
late S we see that starting from t = 0 we have to follow the system during a time
interval τ . For 0 < t < τ there is no entropy and more generally no thermodynamics
since the quantum fluctuations are larger than the standard thermal fluctuations.
383
J.P.Badiali
However, the knowledge of the system evolution during this time interval allows us
to calculate its entropy. For 0 < t < τ we may inspect the system with a time scale
δt � τ . If δt → 0 during the first time interval we have an energy fluctuation ~/2δt
that can be much larger than the mean thermal energy UK. Thus, the evolution of
the system must be such that the fluctuations relax the energy from a high value,
~/2δt, to the mean thermal kinetic energy UK. This is precisely the physical meaning
of (20) which we can rewrite as
UK =
∫
dxa〈uK(xa)〉path =
~
2δt
−
∫
dxa
m
2
〈(δx
δt
)2〉
path
. (27)
Hence, the evolution of the system during the time interval (τ −δt) must be oriented
and cannot result from a time-reversible motion. Of course, after this relaxation the
system recovers its initial equilibrium and there is no net dissipation during the time
interval τ . This can be shown from simple arguments in the case of a free particle.
By analogy with the case of a moving body [19] between two thermal equilibrium
states we can write δS = δE − 〈vδp〉. This expression relates the mean changes
of entropy δS and internal energy δE to the mean change of the mechanical work
〈vδp〉 where v is the velocity and p is the momentum. We may estimate 〈vδp〉 to be
δ〈(1/2mv2)〉 and δE to be δE = ~/2τ = kBT/2. Thus, the absence of dissipation
δS = 0 means that the quantum fluctuation is used to give a mean change of kinetic
energy δ〈(1/2mv2)〉 corresponding exactly to the mean value of the kinetic energy
kBT/2. This shows that the time τ is such that the mean quantum fluctuation is
compensated by the thermal kinetic energy in such a way that there is no entropy
change between equilibrium states.
In summary we have shown that in an equilibrium state the motion is not time-
reversible but appears in a given direction from which we relax the large quantum
fluctuations. We may try to extend our approach in order to describe the irreversible
evolution of a given system which is not in equilibrium at the initial time. This should
lead to a demonstration of a H-theorem. Such a demonstration is possible as it will
be shown in a subsequent paper.
5. Conclusions
In this paper a new derivation of the path integral formalism of statistical me-
chanics is presented. Here we give a physical meaning to the paths. They represent
some motions in a quantized space-time. These motions do not result from a given
Hamiltonian but they are the consequence of the discreteness of the space-time.
Focusing on the counting of paths we recover the results obtained in [10] for the
partition function and the entropy although our results have a different physical
meaning. These results show that the standard thermodynamics is consistent with
the motions that are not time-reversible at a microscopic scale. This offers the pos-
sibility to derive a H-theorem without a new assumption.
The work presented requires future investigations. Many aspects developed here
are taken from the relativity theory. Therefore, it would be nice to elaborate a totally
384
Statistical mechanics in a discrete space-time
covariant theory. In our approach, time-irreversible processes play a dominating role
but it would be important to improve the theory in order to create time-reversible
processes. Some progresses in these two directions have been already done.
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385
J.P.Badiali
Статистична механіка в дискретному просторі-часі.
Термодинаміка і часова необоротність
Ж.П.Бадіалі
Унiверситет iм. П’єра i Марiї Кюрi,
Францiя, 75230 Париж, Площа Жюсс’ї, 4
Отримано 29 квітня 2003 р., в остаточному вигляді – 21 липня
2003 р.
Введення дискретного простору-часу є спробою опису фізики в
масштабі Планка. Нами буде показано, що така концепція може та-
кож бути корисною для опису термодинаміки в пре-релятивістичній
системі. Виходячи з цієї концепції, нами запропоновано новий ста-
тистико-механічний підхід, що базується на динамічній точці зору і
представленні ентропії. Ентропія пов’язана з процедурою розрахун-
ку шляхів в просторі-часі. Вона містить часовий інтервал, що являє
собою час необхідний для загасання квантових флуктуацій і досяг-
нення термічного режиму. Показано, що β~ є цим часом. Матема-
тичні вирази, отримані нами для таких термічних величин як ентро-
пія і вільна енергія, є ідентичними з такими, отриманими з тради-
ційного формалізму інтегрування по траекторіях, починаючи з кано-
нічної форми термічної матриці густини. Проте, введення квантова-
ного простору-часу показує, що термодинаміка узгоджується з рів-
нянням руху, яке є незворотнім в часі на мікроскопічному рівні. Як
наслідок, знову постає проблема необоротності і вивід Н-теореми
стає можливим у майбутньому.
Ключові слова: статистична механіка, термодинаміка,
необоротність часу, дискретний час-простір
PACS: 03.65.Ca, 05.30.-d, 05.70.-a, 47.53.+n
386
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