Probability distribution for a lattice gas model

Physica A 152 (1988) 243-253 North-Holland, Amsterdam

PROBABILITY

DISTRIBUTION

FOR A LATTICE

II. THERMODYNAMIC J.GI%MEZ, Departamento

S. VELASCO

GAS MODEL

LIMIT

and A. CALVO

HERNANDEZ

de Fisicu, Facultad de Ciencias, Universidad de Salamanca, 37008 Salamanca, Spin

Received

9 February

1988

The behavior of the probability distribution for a lattice gas obtained in a preceding paper analyzed in the so-called thermodynamic limit. Two situations, corresponding to cases in which ratio between the subvolume under consideration and the total volume remains finite or tends zero, are discussed. The latter case allows us to write the probability distribution in terms of intensive parameters. The numerical analysis of the probability distribution in this case and implication of the results in order to explain the features of a liquid-vapor phase transition reported.

is the to the the are

1. Introduction In the preceding paper’) (hereafter referred to as paper I), we developed a formalism of conditional probabilities for deriving the probability distribution lV,,(N, V, V,) of a pairwise interacting lattice gas kept at a constant temperature T. This distribution answers the question: Given A’, identical particles enclosed in a volume V,, what is the subvolume V? We have shown that, distribution can be written in the form: W~cj(Nj Vi Vi) = wg:(N,

probability to find N particles in a for our lattice gas model, such a

V; V,,) exp{(alkT)(N

- Nv)2/V(1 - v/v,)}

, (1.1)

where

is the hypergeometric and

distribution

characterizing

the non-interacting

IV” = N,,V/I/;, 037%4371/88/$03.50 0 (North-Holland Physics

lattice

gas

(1.3) Elsevier Science Publishers Publishing Division)

B.V.

244

J. G&MEZ

et al

is the mean value of distribution (1.2). The parameter a, given by eq. (3.4) of paper I, arises from the attractive part of the binary interparticle interaction and (T takes into account the hard-sphere character of the particles. In paper I, a numerical study of distribution (1.1) shows the appearance of bimodal (two-humps) distributions for temperatures below a critical value, That study was made for a relative large number N, of particles. However, macroscopic systems are composed of an extremely large N0 of particles enclosed into a volume V,, which is very large compared to molecular dimensions. In this situation, it is customary to carry out analysis in the so-called thermodynamic limit’):

N,-+m >

v,,+=

7

&IN, = u,): finite constant

(1.4)

In this limit, the overall specific volume u0 stays fixed and remains an important parameter for all physical properties of the system. The purpose of this paper is to analyze the behavior of the probability distribution (1.1) in the limit (1.4). In such an analysis the size of the subvolume V plays a very important role and two situations can be discerned: (i) V is of the order of V, so that V/V, is not too small, or (ii) V is sufficiently small so that V + V,,. The first case is studied in section 2 and it corresponds to the so-called Gaussian approach. The second one is studied in section 3 and it corresponds to the system V immersed in a particle-energy reservoir. This allows us to connect with the grand canonical ensemble and thus to obtain information about the equations of state of the lattice gas model under study. Then, it is possible to write the probability distribution (1.1) as a function of intensive parameters characterizing the reservoir. The numerical study of this function is made in section 4. This study shows the appearance of two-humps distributions for temperatures below the critical value reported in paper I. One shows that the extrema of these distributions reproduce the equations of state of the lattice gas.

2. Gaussian approach When NC)and M - N, (M = Vo/u) are large, the hypergeometric distribution (1.2) presents a sharp maximum at the value (1.3). Then, following a simple method based on the expansion of the neperian logarithm of (1.2) in a Taylor series about #, and truncating this series up to the second order, one finds that distribution (1.2) can be approximated by the Gaussian3,4) W$,‘(N, V; V,) = [2nA&J1’2

exp{ -(N - fiV)2/2A:,,} ,

(2.1)

PROBABILITY

DISTRIBUTION

FOR A LATTICE GAS MODEL II

245

where

A;u,= N,(VII/,)(l

- V/V,)(l-

Noc7/V0)

(2.2)

is the dispersion of the hypergeometric distribution (1.2). An analysis of the higher-order derivatives than the second one of In Wk’(N, V, V,) with respect to N, evaluated at N = sV, shows that expression (2.1) provides a very good approximation to the extent that NOand M - N,, are large and that neither V/V, nor (1 - WV,) are too small. Substitution of (2.1) into (1.1) leads to the Gaussian distribution: WNO(N,V; V,) = [27rA2]-“2 exp{ -(N - NV)‘/2A2} ,

(2.3)

where A2 = N,(VIV,)(l x [l-

- V/V,)(l

- N,cT/V~)

(2N,,lVo)(l - N,dV,)(a/kT)]-’

,

(2.4)

i.e., for a very large number of particles, the probability distribution characterizing the lattice gas model under study has a Gaussian form, centered about &‘,, with a temperature-dependent dispersion given by (2.4). Although the Gaussian approach leads to an explicit expression for the dispersion, distribution (2.3) contains less information than the initial distribution (1.1) about the behavior of the gas. In particular, distribution (2.3) cannot present a bimodal form and does not give information about the possibility of two coexisting phases or the existence of metastable states. However, dispersion (2.4) carries information about density fluctuations for our lattice gas model. From this expression one gets that for the temperature T = (2alk)(u,

- CT)/u;

(2.5)

the density fluctuations become infinite. For temperatures above (2.5), dispersion (2.4) is positive, NV is the only maximum of distribution (2.3) and so the homogeneous density N,lV= N,,IV, is a stable state of the system. For temperatures below (2.5), dispersion (2.4) is negative and distribution (2.3) has no probabilistic meaning, but, from a formal point of view, NV is a minimum of function (2.5) and the homogeneous density can be considered as an unstable state. Eq. (2.5) is precisely the instability curve obtained in paper I. Apart from the above considerations, two interesting relations between dispersion (2.4) and the corresponding to the non-interacting lattice gas and the ideal gas should be remarked:

246

J. GfkMEZ

i) Comparison between above (2.5) one has A’ > Af a i.e.,

)

and

(2.4)

shows

that

for

any

temperature

(24

’

the attractive

the density

(2.2)

et al.

part

of the interparticle

potential

leads to an increasing

of

fluctuations.

ii) For T = (Zalkc~)(u, dispersion

(2.4)

- 0)/u,,

(2.7)

becomes

A’ = A;,,,N,,(V/v,,)(l

- V/v,,) .

(23)

This is the familiar dispersion for the binomial distribution characterizing the ideal gas. If the gas density is low enough such that u,, is large compared to the volume cr of the hard core of the particles, temperature (2.7) reduces to the well-known Boyles temperature (T,, = 2aik~) for the van der Waals gas.

3. Grand canonical

approach

Now we consider that in the thermodynamic limit process the size of the subvolume V remains fixed so that Vi& tends to zero. Then, dispersion (2.2) also tends to zero and truncation of the Taylor expansion of In W$j up to the second order is not justified. In order to analyze the behavior of (1.1) in this case, we write the hypergeometric

distribution

(1.2)

in the form

(3.1) K!(M Then,

using

the approximation

N,!l(N,-N,)!=NY and taking

- K)!

into account

(N,;N,-&al)

(3.2)

that

No/M = N,,uIV,, = u/u,, = @,I K

(3.3)

PROBABILITY

DISTRIBUTION

remains finite in the thermodynamic

247

FOR A LATTICE GAS MODEL 11

limit, expression

(3.1) takes the form (3.4)

Expression

(3.4) may now be substituted

W(N, V) = (E)(

$,“C$J””

into (1.1). Since V/V,,+O, exp((alkT)(N-

NJ21Ko-} ,

we get (3.5)

where K = VI(T and i?, is given by (1.3). The size of subvolume V is kept constant in the above limit process. This means that we treat a finite system (V) in equilibrium with a particle-energy reservoir (V,). For this reason, we refer to (3.5) as the grand canonical approach of distribution (1.1). In this context, one easily checks that in the gas ideal limit (a+ 0, cr+O), (3.5) becomes the well-known Poisson distribution: W,,(N, V) = e-‘v-N,IN!

(3.6)

In order to establish a connection between our probabilistic analysis and the thermodynamics of the system under study, it seems suitable to write distribution (3.5) in terms of intensive parameters characterizing the reservoir. With this aim, we write distribution (1.1) in the form

(3.7) On the other hand, since the given system, which is composed of N,, identicle particles enclosed into a volume V,, can be considered as a member of a canonical ensemble characterized by temperature T, probability distribution WNo(N, V; V,) is given by5) Wy,(N, v; v,) = G(T,

V)Z,_,(T,

V, - V)iZ,,JT,

where Z,“(T, V,) is the canonical partition function Comparison between (3.7) and (3.8) then gives

v,) ,

(3.8)

of the total system.

J. G6kMEZ

248

where

the dimensionless

factor

a = 0, with the exact partition

et al.

y can be derived function

by comparison

of a non-interacting

of (3.9),

lattice

for

gas6):

(3.10) with M = V,,/a and h the so-called y = Next,

from

thermal

wavelength.

crlh3.

Thus

one gets (3.11)

by defining

(3.9)

the pressure

and the chemical

potential

of the reservoir

by

we obtain

P,IkT = -( 1 I(T) ln( 1 - a/u,,) -p”/kT which

mean

= In y + ln(v,/p

are the well-known

- (alkT)(

- 1) + (2a/kT)(l

equations

of state

1 /ui)

,

(3.12b)

iv,) , of a lattice

(3.13b) gas with an attractive

interaction -2alu, between each pair of particles7). Then, taking into account (3.7), in view of eqs. (3.12b) and (3.13b), distribution (3.5) may be written as W(N, V) =

cJ

yN

K N

exp{ -P,VlkT

+ pL,,NIkT + (alkT)N21V}

(3.14)

Although, for the derivation of (3.14), we have assumed a system with a fixed V in equilibrium with a particle reservoir (and so as a member of a grand-canonical ensemble), expression (3.14) can be also formally interpreted as a probability distribution for the variable V. This situation should correspond to a system with a fixed number of particles N in equilibrium with a volume reservoir (and so as a member of an isothermal-isobaric ensemble).

4. Numerical

analysis

In this section we shall make a numerical analysis of distribution (3.14), in order to investigate the possibility as to whether such a distribution could

PROBABILITY

DISTRIBUTION

FOR A LATTICE GAS MODEL II

249

present a bimodal form. This analysis can be carried out by assuming (3.14) as a distribution for N by a fixed V value, or, alternatively, by assuming it as a distribution for V by a fixed N value. In the first case, the temperature T and the chemical potential pO of the reservoir play the role of controlable parameters, while, in the second case, such a role is played by the temperature T and the pressure PO. Since the last case is the most usual macroscopic situation, we shall consider (3.14) as a distribution for V: WV)

m (E)

PVIkT + (alkT)N2/V}

exp{-

,

(4.1)

where P denotes the pressure of the system (the system and the reservoir are in a state of mutual equilibrium, and so they have a common pressure PO = P). By introducing the reduced variables (dimensionless) V* = V/CT= K ,

expression

T* = (kula) T ,

P* = (a2/a)P,

(4.2)

(4.1) becomes

W(V*)a(

P*V*IT*

y>exp{-

+ N=IT*V*)

.

(4.3)

By taking the Neperian logarithm of (4.3), using the Stirling approximation and then derivating with respect to V”, one gets (a In W(V*)ld

V*),.,,

= In V* - In(V* - N) - P*IT*

- N21T*V*‘. (4.4)

So, the extrema P*IT*

V* of (4.3) verify the equation

= -ln(l

- l/V*) - 1/T*rJ*2,

with U* = *IN. Eq. (4.5) coincides with (3.12b) derivating (4.4) respect V* one obtains 8’ In W(V*)18V*2

and so the character T* >2(U*

(4.5)

for rJ* = ug. Next,

= 1 /V* - 1 l(V* - N) + 2N*/T*V*“,

by

(4.6)

(maximum or minimum) of V* is given by

- l)/~?*~-+ U*

T” < 2(U* - 1)/V*2*

is a maximum of (4.3),

(4.7a)

V* is a minimum of (4.3))

(4.7b)

250

i.e.,

J. G&ME2

et al.

the curve T”

delimits

= 2(c*

-

1)

/c*”

(4.8)

in the (U*, T*) plane, according to (4.7), the maxima (stable states) (unstable states) of distribution (4.3). Eq. (4.8) coincides with

and the minima

(2.5) for ii* = uz, and gives equation presents a maximum

-* v,-2,

the instability at

curve

obtained

= 0.5 , Pr=$(2ln2-1).

Tr

in paper

1. This

(4.9)

Eq. (4.8) and the point (4.9) have a direct thermodynamic interpretation: one checks easily that (4.8) have a direct thermodynamic interpretation: one checks easily that

(a P"

(4.8)

is the spinodal

/a u*)T*

of the equation

curve

= 0

of state

(4.10) (4.5),

and (4.9)

is its critical

point.

To investigate the influence of T* and P” on the behavior of distribution (4.3), we have carried out a numerical analysis of expression (4.3) for N = 1000 and various values of T* and P” around the critical point (4.9). For T” = 0.52, 0.5 and 0.48 we obtain the results shown in fig. la, b and c, respectively, where we have also plotted the location 6” of the extrema of (4.3) as a function of the reduced pressure P*. These results can be summarized as follows: (4.3) presents a i) For T* >0.5 (fig. la) and any value of P*, distribution single peak (stable state) whose abscission moves towards the region of small molecular volume (high density) with a decrease of the width as P” increases. (ii) For T” = 0.5 (fig. lb) and P* < a(2 In 2 - l), distribution (4.3) presents a single peak (stable state) located at V* > 2. The width of this peak increases as P* increases, so that for P* = $(2 In 2 - 1) distribution (4.3) presents a plateau centered at U* = 2 (critical point). For P* > 4 (2 In 2 - 1) distribution (4.3) presents again a single peak (stable state) located at U* < 2 with a width rapidly decreasing as P* increases. iii) For T” < 0.5 (fig. lc) and low enough P” values, distribution (4.3) presents a single peak (stable state) with an abscission located in the region of high molecular volumes (small densities). By increasing P*, distribution (4.3) has a bimodal form with two peaks, one of them located in the region of high molecular volumes (>2) and another one located in the region of small molecular volumes (<2), so that the height of the first peak (stable state) is higher than the second one (metastable state). Between the above two peaks a minimum (unstable state) is located with an abscission given by the homoge-

PROBABILITY

DISTRIBUTION

FOR

A LATTICE

GAS

1

-h * >

MODEL

II

251

0.08

0.088

s

1

0

5*

6

0

P’

6

Fig. 1. Behavior of probability distribution (4.3) versus u* = V*IN for different (T*, P*) values: a) T* = 0.52, b) 0.5, c) 0.48. (In all cases N = 1000). The lower diagrams show the dependence of the extrema U’ of the distributions on the reduced pressure: absolute maxima (0), relative maxima (O), minima (A). (In these diagrams, the corresponding isotherms of the equation of state (4.5) are also plotted.)

neous density. By further increasing of P*, the two peaks become more equal in height, so that for a given P* value both have the same height (probability). Again increasing P*, distribution (4.3) is still bimodal, but now the absolute maximum (stable state) is located in the region of small molecular volumes (~2) while the relative maximum (metastable state) is located in the region of the high molecular volumes (>2). For high enough P* values, distribution (4.3) presents again a single peak (stable state) in the region of small molecular

252

J. GokMEZ

et al

T' 0.51 0.50 0.49 0.48 0.47 0.061 0

1

2

ij*

4

3

I 5

Fig. 2. Extrema of the probability distribution (4.3) versus P* for different T* values. denote absolute maxima, squares relative maxima, triangles minima. Solid lines correspond physical isotherms.

volumes (high densities) values of P*.

with a width

smaller

than

that corresponding

Summarizing, the above numerical study of distribution lattice gas model under study, the following: 1) The existence of a single single peak of the distribution. 2) The existence of a critical

stable point

phase

for

(4.3) reveals,

T* > 0.5,characterized

with coordinates

given by (4.9),

Circles to the

to low

for the

by a charac-

terized by a distribution with a plateau centered about 6: = 2. 3) The existence of two stable phases for T" ~0.5,one of them at low densities (vapour phase) and another one at high densities (liquid phase). 4) The existence of metastable states for T* < 0.5, characterized by relative maxima of the distribution. 5) The possibility of coexistence of both phases for a P* value for which the two peaks of distribution (4.3) have the same height. 6) The characterization of the liquid and vapor phase not only by the location of the peaks but also by the width of the peaks: the peaks correspond-

PROBABILITY

DISTRIBUTION

FOR

A LATTICE

GAS

MODEL

II

253

ing to the liquid phase are sharper (less dispersion and thus less density fluctuations) than the peaks associated to the vapor phase. The results of the dependence of the extrema V* of distribution (4.8) on T* and P* are plotted in fig. 2. In this figure the numerical results are plotted by symbols: absolute maxima (0) relative maxima (Cl) and minima (A). This representation allows a clear probabilistic interpretation of the states given by the equation of state (4.5). This fact allows us to plot the “physical isotherms” from the following of the absolute maxima of distribution (4.3). For T* < 0.5, such isotherms present a plateau just for the P* value for which the two peaks have the same height. One can check that such a line divides the “loop” displayed by the theoretical isotherm in two regions of equal area, i.e., the well-known Maxwell equal area rule is interpreted in our probabilistic formalism as the equal height rule for the two peaks of distribution (4.3).

References 1) J. Giitmez, S. Velasco and A. Calvo Hernandez, Physica A 152 (1988) 226, paper volume. 2) R. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley, New York, 3) W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New 1968). 4) F. Reif, Fundamentals of Statistical and Thermal Physics, (McGraw-Hill, New York, 5) K. Huang, Statistical Mechanics (Wiley, New York, 1963). 6) J.M. Pimbley, Am. J. Phys. 54 (1986) 54. 7) P.T. Landsberg, Problems in Thermodynamics and Statistical Physics (Pion, London,

I, this 1975). York, 1965).

1971).