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Unusual statistics for extended systems
Stefano Isola, Stefano Ruffo
To cite this version:
Stefano Isola, Stefano Ruffo. Unusual statistics for extended systems. Journal de Physique II, EDP Sciences, 1991, 1 (11), pp.1349-1361. �10.1051/jp2:1991144�. �jpa-00247596�
Classification Physics Abstracts
05.45 05.20 05.50
Unusual statistics for extended systems
Stefano Isola (',*) and Stefano Ruffo (2.*)
I') Dipartimento di Matematica e Fisica, Universiti di Camerino, 1-62032 Italy
(~) Dipartirnento di Clfimica, Universiti della Basilicata, Potenza, Italy and INFN, Sezione di Firenze, Italy
(Received 8 January J99J, revihed 29 May J99J, accepted 26 July J99J)
Rksulnk. On donne un exemple de mod61e statistique d6crivant l'dchange d'dnergie pawlfi les modes propres d'un systdme physique assez gdndrique. Les propridtds thermodynamiques sont traitdes de fagon ddtaillde, aussi bien que certaines gkndralisations. On montre que la densitE
d'entropie est inddpendante de l'existence de lois de conservation de l'dnergie. Cela rend le rdsultat applicable au cas conservatif aussi bien que dissipatif. La distribution la plus probable a dtk calculde en utilisant la mdthode de Darwin-Fowler. On donne aussi les corrections de taille
finie. Enfin, on donne une application de la formule pour la densitd d'entropie aux donnes
expdrimentales ddcrivants les fluctuations d'amplitude des modes propres dans la convection de
Rayleigh-Bdnard dans un anneau.
Allstract.- We give an example of a statistical model which describes the exchange of energy among the normal modes of a generally designed physical system. The thermodynamic properties
are computed in detail, as well as some generalizations. The main result concems the
independence of the entropy density on the presence of energy conservation laws. Tiffs makes the result applicable to both conservative and dissipative cases. The most probable distribution is
computed using the Darwin-fowler method. Finite volume corrections are also given. An
Appendix is devoted to the use of our formula for the entropy density to analyze experimental data on mode amplitude fluctuations in Rayleigh-Benard convection in an annulus.
t. Introduction.
In many physical situations one has to do vith a large collection of objects exhibiting a statistical behaviour. Consider for instance a one-dimensional system of N oscillators
interacting via a non-linear coupling between neighbours. If the total energy E is large enough, the N quasi-particles corresponding to the Fourier modes exchange their energy in a random fashion, so that any initial condition leads in a short time to an equipartition state [I].
Tliis means that the time average (E,) of the energy contained in the I-th mode, for
I
= I,
,
N, is a constant equal to E/N. It is reasonable to expect, and also veRfied for some
specific models [I], that mode amplitudes show Gaussian fluctuations around this average value with a given variance ~,, which in turn can be a function of the energy and of the
considered mode.
(*) Mail address INFN, Sezione di Firenze, L. go E. Fernli 2, 50125 Firenze, Italy.
Similar situations arise in experiments on spatio-temporal chaos [2]. Temperature signals
are monitored in different points of a quasi-one-dimensional Rayleigh-Benard convection cell and Fourier analysis is performed. In the state of fully developed space-time chaos the modes exchange energy at random and the amplitude distribution is Gaussian in each mode with a
variance depending on the mode. The total energy E, defined as the sum of squared Fourier amplitudes, is not conserved, but fluctuates at random around a well defined mean value (see Fig. 3 and the Appendix).
These two physical situations, among others, have basically motivated the construction of the following statistical model.
Let us consider a collection of N objects, called modes, or oscillators, for definiteness, though they could be any other physically identifiable objects. A total energy E is at their
disposals and it can be exchanged in quanta of size e, so that the whole system of modes has
n = Ele quanta. We know that in the above described physical contexts a configuration where most of the energy is concentrated in one mode is rare, e.g. its probability is given by some point in the tail of a gaussian. Thus, in our model we impose the constraint that the energies
(E,) of the modes take values in the interval if &, I+ &], where the mean energy li
= E/N
= ne/N
= ue is a multiple of the basic quantum e, and u is the mean number of quanta per oscillator. For the sake of simplicity we also require that each energy state has the
same probability. In other terms, we consider a flat density of states p (E, )
= I with support
on the interval above, where the width of the distribution is also measured as
= de, such
that the number of states in each mode is 2 A + I. Of course, this is a crude approximation of the gaussian case, but we shall see that already this simple model shows interesting features.
Our main goal will be the computation of two quantities: the entropy density in the
thermodynamic limit. N
- oJ, n - oJ with n/N
= const, as a function of u and A, and the most probable distribution of the quanta among the modes.
This statistical model introduces an ensemble which is in some sense intermediate between the Bose-Einstein and the Fermi-Dirac cases. However, these two statistics cannot be
reobtained as particular limits of our ensemble, as for instance happens for the intermediate statistics first introduced in reference [3]. This is mainly due to the constraints that we impose and, secondarily, to the energy spectrum that we choose. The relaxation of these constraints and the choice of a different spectrum are interesting modifications of our model.
This paper is organized as follows. In section 2 we report the calculation of the entropy in the thermodynamic limit, together with some results concerning a gaussian density of states and the Bose-Einstein statistics. In section 3 we derive the most probable distribution using a
method due to Darwin and Fowler. In the same section we discuss some numerical results which show the rate of convergence to the thermodynamic limit. Section 4 is devoted to some conclusions and perspectives. In the Appendix we present an application of our formula for the entropy density to an experiment on Rayleigh-Benard convection.
2. En«opy.
The counting procedure is different from the one leading to the Bose statistics. In that case one begins with the state where all the quanta are given to one oscillator, this case being
realized in N ways, then one generates a new state with degeneracy N IN I by subtracting
one quantum and assigning it to any other oscillator, and so on. This procedure reduces the calculation of the volume to that of the permutations of N + n objects.
In our case it is preferable to begin with the equipartition state and then count the number of ways one can subtract and reassign quanta to the oscillators. This makes the problem
straightforwardly related to partitions of integers. Let us start with a simple example. Let us
consider the case in which we have four oscillators, N
= 4. The number of quanta is
n = 24 so that the mean number of quanta per oscillator is u
= n/N
= 6. Let us fix to
A
=
2 the range of variability of the energy density u,, for I = I,..,N. The allowed
configurations are reported in table I, with their multiplicities. The equipartition state is the
one where the I-th oscillator has u, = u = 6, and a new state is generated by subtracting one quantum -to one of them and reassigning it to another this can be done in N II (N 2)
= 12
ways (see the second line of Tab. I). The next class is obtained by subtracting two quanta this can be done in two ways, the quanta can be deducted either both to a single oscillator or
one to two distinct oscillators in the process of reassignement again one can choose the two possibilities. Thus the number of possibilities is counted by the partitions of 2 as the sum of two non negative integers : 2
= 2 + 0
= + 1.
Table I. Configurations of the ensemble of modes with their multiplicity when
N
= 4, n
= 24, A
= 2, u
= 6.
Configurations Multiplicity
6 6 6 6 41/4!
5 7 6 6 41/2! 12
5 5 7 7 4!/(21)2 6
5 5 8 6 41/2! 12
4 8 6 6 4!/2! 12
4 7 7 6 4!/2! 12
4 5 8 7 4! 24
4 4 8 8 4 !/(2 !)2 6
This procedure extends directly to cases where more than two quanta are deducted, involving the knowledge of the number of ways a positive number m can be written as the sum of k integers. Unfortunately this number is not known in general. Moreover, following this
procedure, one has to take into account explicitely the bounds imposed by the finiteness of A and by the total number of oscillators N. In our example one cannot subtract or give more
than two quanta to each oscillator and, at the same time, one cannot divide them in a number of parts that exceed the number N. For instance, the partition 3
= 2 +1 is allowed, but
3 =1+1+1 is not.
From table I the energy distribution can be easily obtained and it is reported in figure I.
One can repeat the exercise for different values of N, n and A and obtain results with the same features ; the energy distribution has support on the interval iv A, u + A and is symmetric
around u.
A case where the calculation of the volume (the total number of allowed configurations)
can be carried out is that with A
= I. The configuration obtained from the equipartition state
by'subtracting one quantum and giving it to another oscillator, I.e.
u- I,u+ I,u,...,u
(2.I)
N has multiplicity N !/(N 2 The next one
u-I,u- I,u+ I,u+ I,u,...,u
~ .~ (2.2)
N
O.25
O.20 ~ ~ ~~~
x x
o.15
o.io
o.05
o.oo
2 4 6 8 10
O.125
P 16)
O.100
~ ~ ~ ~ ~
~ x x
x ~ x
0.075
o.oso
O.025
u
o.coo
lo 15 20 25 30
Fig. I. Normalized energy distribution P(u,) for two different sets of values of the parameters.
N is the number of modes, 4 the width of the single mode distribution and u the mean number of quanta per mode. The results do not depend on the number of quanta n. (a) N
= 4, 4 = 2, u
= 6; (b)
N =10, 4
= 5, u
= 20.
has multiplicity N !/(N 4)1 (2 !)2. The process can be continued until the number of subtracted quanta reaches N/2. Therefore the volume fl is given by
fl
= £ ~
~
(2.3)
~~(N-2k)!
(k!)
At this point it is crucial to note that in the general case the volume can be computed by exploiting an analogy with spin systems. Looking for instance at the configurations (2.I) and
(2.2) it is evident that subtracting everywhere u they reduce to the configurations of a three
states (-1, 0,1) spin system, where the total magnetization is fixed equal to zero.
In general we are dealing with a (2 A + I )-state spin system with zero magnetization.
Hence the microcanonical volume is
JV
fl
= £ £ uj~~ uN (2.4)
fE cant k
where the index (I) labels the distinct (2 A + I configurations over which the sum extends,
and is the Kronecker symbol : (0)
=
and (x)
= 0 for x # 0.
In order to perform the sum in (2.4) we choose the Fourier representation of the Kronecker
&-symbol
(x)
=
£ le'P~ dp (2.5)
so that, after exchanging the integral with the sum we obtain fl
= l~ dp e 'P~~ £ exp ip (
uj~~ (2.6)
~ ~
« fE cant k
Since we are dealing with non interacting spins, the above sums factoRze into the
contributions pertaining to each site : fl
= l~ dp e ~'P~~ jj £
exp (ipuj~~)
2 w
~ ~
j « J N
= dp 2
£ cos ~pi + (2.7)
~"
-« i=1
Notice that the volume does not depend on u. Tile integrand in (2.7) is a 2 w-periodic
function of p vlith maximum in p = 0. Thus, in the thermodynamic limit N
- oJ, the saddle
point method gives the entropy density
s = lim ~°§~ = log (2 A + 1) (2.8)
N
- w
Let us observe that in the thermodynamic limit the constraint on the total number of available quanta is not effective. In other words the entropy density is simply the logarithm of the number of states per site. This is physically understandable since :
(= ~~
= 0 (2.9)
and therefore the temperature of our ensemble is infinite.
For the practical use of these formulae it may be useful to compute corrections to the saddle point. Corrections due to the presence of other extrema ~p # 0) are exponentially small
O(exp(- N)/N). The second order correction to the dominant saddle point in p
= 0 is
~~ /f ~°~ (A~+
I l~~~j~i~ ~~'~~~
It may be of some interest to point out that our derivation provides incidentally the identity
~~
(N 2~~ jk )2 ~ j~ ~~
~~ ~°~~ ~ ~~ ~~'~~)
obtained from (2.3) and (2.7) for A
=
1.
2.I THE GAussiAN cAsE.- It may be of some interest to derive an expression for the
entropy in a case where the states are gaussian distributed. This means that the density of states with energy E, is given as
p (E,)
= exp i- (E, t)2 «/~/i (2-12)
with the constraint that the total energy is fixed to E. The Gaussian in (2.12) has been
normalized in such a way that the number of states in the I-th mode is ~,, with
~, m I (we allow that this number depends on the mode). Now the volume is given by :
+W. +W
IN
N
fl
= £ E, E fl dE, p (E;) (2.13)
-w -w ,=i ,=1
where &(x) is the Dirac 3-function. After choosing the Fourier representation of the 3- function one can rewrite the volume in terms of gaussian integrals
+W .A~ +W
fl ~
= dp fl dE, exp ipE,
~E)~ (2,14)
2 "
w , i w ~,
By shifting the integration path in E; to an axis parallel to the real axis one obtains :
flN ~,
fl
=
'"~ (2.15)
( ~)
'
from which the entropy density at finite volume :
N N
£ log ~, log jj ~/
s~ =
' ' (2.16)
In the thermodynamic limit N
- oJ, the second term, which arises from the conservation law, vanishes at least as log (~$~~ N)/N, whereas the first term gives the entropy density as
the average of the logarithm of the number of states per mode :
£N log ~,
s= Em '
~ = (log~,). (2.17)
N - w
This formula has been used to fit the experimental data of reference [2], following the
dependence of s on a control parameter given by the temperature difference between the
plates in the convection cell. The result are in nice agreement with those obtained using the
spectral entropy published in [2].
It is important to stress that also in this case the result is insensitive to the constraint of
fixing the energy, thus showing its applicability to both conservative and dissipative cases. Tile
only difference for the dissipative case is the fact that N is not in general the dimension of the space, but that of the attractor, which can be smaller.
2.2 THE BosE-EINSTEIN cAsE. It is also possible to relate the,counting used in this section to that used in the Bose-Einstein case. Again one has N objects and n quanta, but now the number of quanta in each mode can vary in the interval [0, n]. We have thus to do with an
(n + I )-state spin system, with the constraint that the total magnetization is n. We can thus rewrite the Bose-Einstein volume as
fl = jj I(
w, n (2.18)
w,j ,=1
where w, denotes the spin variable corresponding to the I-th mode.
Using again the Fourier representation of the Kronecker &-symbol, equation (2.5), one
obtains the following identity :
The saddle point solution of the integral in the I-h-s- gives the Stirling approximation of the
expression in the r-h-s-
3. The most probable distribution.
We wish to find the most probable distribution of energies among the N modes, in the limit as
N-oJ. Putting for simplicity e
= I, this amounts to compute the distribution of
N particles in 2 A + I cells, each of them bearing an energy E~
= u + k, where A w k « A
and u is the mean energy per particle. Let us denote by n~ the number of particles occupying
the k-th cell. Then, the constraints we are imposing can be expressed by
J d
£ (u+k)n~=Nu and £ n~=N. (3.I)
k=-J k=-J
It is straightforward to realize that (3.I) implies
£J kn~=0 (3.2)
k=-J
and, since (3.2) holds for any value of A, we get
~k " ~-k (3.3)
that is, the distribution is symmetric about the average energy u. Furthernlore, from (3.I) we easily obtain that the n=~ s range over the set (0,1,.., n (]~~~) where
We point out that this expression can also be useful to clarify differences and analogies
between our distribution and both the usual Fermi-Dirac and Bose-Einstein quantum
statistics, and the intermediate statistics obtained in [3].
In order to calculate the most probable distribution, we apply a method put forward by
Darwin and Fowler (see, for instance, [4]). Let us first give a rapid outline of this method. The main point is to find the most probable set (i~) by calculating the value of n~ averaged over
all possible distributions in energy compatible with (3.I), together with the mean square fluctuations (n() (n~)~. If the latter vanish in the limit N
- oJ, then in that limit :
(n~) -i~. On the other hand these quantities can be easily expressed in terms of
log fl where fl is the number of states of the system. Thus, one has first to compute
fl as a function of N and u.
In our case the sequence (E~) can be assumed to be a sequence of non-decreasing integers
without loss oi generality. A generating function can be defined as
f(z)
= £ z~~fl (N, r) (3.5)
JOUR~AL DE PHYSIQ~E II T I, V II ~OVE~BRE 199J
where the sum extends over all rational numbers r varying between u A and u + A with steps of size 2 A/N and such that Nr is an integer (u and N being kept fixed). The calculation of the volume fl must take into account the constraints (3.I).
Since fl(N,u) is the coefficient of z~~ in the expansion of f(z) in powers of
z, one can obtain the volume fl integrating in the complex plane around the origin
z = 0,
niN, u)
=
i i'i±~> d~
=
~~ fi~~'~
° ~~~~
gives the number of states corresponding to any given u and N.
Now, since log f(x)
=
O(N) and N » I, we can evaluate the integral (3.6) as follows. Let
g(z) be defined by
g(z)
= log f (z) Nu log z (3.7)
Giving the point J~ (usually real and positive, as it is in our case) is determined by
g,jx),~
~ ~ =
o (3.8)
and we integrate (3.6) along the circle (z =xoe"<0w~§ w2w). The integration can therefore be approximated near this point, where the integrand has a sham maximum
fl (N, u
= j~
~
exp (g(J~) +
~~~'~~
(J~ e'~ x~)2 + d~§
2 "
o 2
m
) j~
~
eXP
g(~b)
~~~~xi~b
) d~§ (3.9)
o
~
i ~giJ~)
J+ ~
and
log n g (~) log (2 «xi g"(~)) (3, lo)
where the second term is O(logN).
In our case, the function f(z) in (3.5) can be rewritten in the following form
J
~
N A N
f(z)= k=-d£ z ~ =z~~ k=-J£ z~ (3.ll)
where we have simplified the sums and the constraints still present in (3.5).
Therefore,
IA
g(z)
=
N log £ ~)
(3.12)
k A
and equation (3.8) gives :
jjJ kz~
= 0. (3.13)
k=-J
