statistical thermodynamics of money (thermoney)statistical thermodynamics, like entropy into money....

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 5 (2016) pp 3409-3420

© Research India Publications. http://www.ripublication.com

3409

Statistical Thermodynamics of Money (Thermoney)

Dr. S Prabakaran

Associate Professor School of Economics & Business Administration, Department of Accounting & Finance, Pondificia Universidad Javeriana Cali. Cali, Colombia.

Abstract

Thermodynamics presents an abstract and generalized

approach that enables one to analyze the basic regularities of

various energy processes, even under conditions where the

details of their intrinsic mechanisms are unknown. The

methods of thermodynamics are applicable to systems that

belong to very diverse classes of objects from starts to living

cells. In this work, we attempt to construct the bridge between

statistical thermodynamics and financial side. The main

purpose of this paper to introduce concepts borrowed from

statistical thermodynamics, like entropy into money. The main

goal of this study is fourfold:

1) First we begin our approach through the analogy relation

between statistical thermodynamics and economics

system.

2) Next we demonstrate how the Boltzmann – Gibbs

distribution emerges in economics models.

3) Next we extend this approach through the hypothetical

economic systems in a linear and nonlinear system

4) finally; we construct the thermodynamics monetary system

at Polytropic Constant ( tConsVP nN tan .

and this paper end with conclusion.

Keywords: Statistical Mechanics, Thermodynamic, Money

and Boltzmann – Gibbs distribution, Polytropic Constant.

Historical Introduction The application of statistical physics methods to economics

promises fresh insights into problems traditionally not

associated with physics [1]. Both statistical mechanics and

economics study big ensembles: collections of atoms or

economic agents, respectively.

The fundamental law of equilibrium statistical mechanics is

the Boltzmann-Gibbs law, which states that the probability

distribution of energy is

TCeP

(1)

Where T is the temperature and C is a normalizing

constant [2].

The main ingredient that is essential for the textbook

derivation of the Boltzmann-Gibbs law [2] is the conservation

of energy [3]. Thus, one may generalize that any conserved

quantity in a big statistical system should have an exponential

probability distribution in equilibrium.

An example of such an unconventional Boltzmann-Gibbs law

is the probability distribution of forces experienced by the

beads in a cylinder pressed with an external force [4]. Because

the system is at rest, the total force along the cylinder axis

experienced by each layer of granules is constant and is

randomly distributed among the individual beads. Thus the

conditions are satisfied for the applicability of the Boltzmann-

Gibbs law to the force, rather than energy, and it was indeed

found experimentally [4].

We claim that, in a closed economic system, the total amount

of money is conserved. Thus the equilibrium probability

distribution of money mP should follow the Boltzmann-

Gibbs law Tm

CemP . Here m is money, and T is an

effective temperature equal to the average amount of money

per economic agent. The conservation law of money [5]

reflects their fundamental property that, unlike material

wealth, money (more precisely the fiat, "paper" money) is not

allowed to be manufactured by regular economic agents, but

can only be transferred between agents. Our approach here is

very similar to that of Ispolatovet al. [6]. However, they

considered only models with broken time-reversal symmetry,

for which the Boltzmann-Gibbs law typically does not hold. It

is tempting to identify the money distribution mP with the

distribution of wealth [6]. However, money is only one part of

wealth, the other part being material wealth. Material products

have no conservation law: They can be manufactured,

destroyed, consumed, etc. Moreover, the monetary value of a

material product (the price) is not constant. The same applies

to stocks, which economics textbooks explicitly exclude from

the dentition of money [7]. So, in general, we do not expect

the Boltzmann-Gibbs law for the distribution of wealth. Some

authors believe that wealth is distributed according to a power

law (Pareto-Zipf), which originates from a multiplicative

random process [8]. Such a process may reflect, among other

things, the fluctuations of prices needed to evaluate the

monetary value of material wealth.

Statistical Thermodynamics Examples of economic systems of interest are an individual

consumer or a small country, each of which is embedded

within a larger economic system. Consider an individual

consumer. A fundamental assumption in economics is that the

consumer employs a utility function U to choose to purchase

one good over another. For many purposes, it is sufficient for

the utility to be an ordinal quantity (that is, it specifies only

relative ordering). However, to make the full analogy to

thermodynamics, we must take the utility U to be a real

number. We assume U to be given in a convenient set of

units, such as 1998 dollars, and we also assume that U is

mailto:[email protected]



3410

measurable [9]. The formalism we develop is falsifiable, and

can be over determined by a proper set of measurements, thus

providing constraints on its consistency. To explain the

analogy, we begin by discussing certain fundamental relations

in economics.

First consider the Measurable Economic Quantity Known as

Wealth,

pNMW (Economics)

(2)

Where and M represents the value and amount of money,

and p and M represents vectors of prices and numbers of

goods.

Economics assumes that the value of an individual

consumer´s money and goods is summarized by the value of

U , which typically exceeds W . The excess is known as the

surplus for which we introduce the notation (psurplus) [10,

11]. Thus

WU (Economics) (3)

In a primitive or very poor economy, there is no surplus, so

0 . In this case, every individual performs the same

economic function at the same efficiency, and there is no

benefit from specialization and trade. The surplus cannot

be negative; for typical economic systems 0 . Although

Equation (3) appears only to define another unknown

quantity, in terms ofU , this economic relationship is

useful because it has a thermodynamic analogue.

The Helmholtz free energy of a system with N identical

particles is defined as [12]

NPVF (Thermodynamics)

(4)

Where P is the pressure, V is the volume, and is the

chemical potential of the particles. We may think of PV

as analogous to M . The quantity in thermodynamics

analogous to the price p is the chemical potential . The

energy E is related to F in terms of the temperature T and

entropy S via

FETS (Thermodynamics)

(5)

Note that, according to the third law of thermodynamics,

0S for a system at 0T

A comparison of Equation (3) and (5) suggests another

analogy, that of handTS . By taking a system with zero

surpluses (and thus zero economic temperature) to have zero

economic entropy, we assume the economic analogue of the

third law of thermodynamics. Here we summarize the

thermodynamic and economic analogies in Table 1.

Table 1: Summary of the suggested analogies between

thermodynamic and economic systems.

THERMO

DYNAMICS F E TS N

ECONOMICS W(Wealth)

U(Utility)

(Surplus)

P(Price)

N (#of

Goods)

Our goal in making an analogy between economics and

thermodynamics is to provide a theoretical framework so that

economics measurements can determine the functional

dependence of the utility U on the economic parameters that

specify the state of an economic system. A knowledge of the

state function as a function of the appropriate economic

parameters completely characterizes the economic system.

From the economic relations we introduce

TS

(6)

And pNMTSWTSU (7)

For E suggests that, from the point of view of its natural set

of variables, we have

NMSUU ,, (8)

Relation (8) is our fundamental assumption.

The economic equivalent is

pdNdMTdSdU (9)

Where

MSNSNM NUp

UU

SUT

,,,

,,

(10)

Let us now apply this theoretical structure

In the Wealth of Nations, Adam Smith distinguishes between

two measures of utility [13]. One measure is the ‘‘value in

exchange.’’ In economics it is conventional to identify the

value in exchange with the price p . From (9), we take this

measure to be the marginal utility per good dNdU at fixed

S and M . Another measure is the ‘‘value in use, ’’ which is

less readily identified. We will identify ‘‘value in use’’ with

the marginal utility per good dNdU for another set of

fixed variables. For simplicity, we will take M to be fixed,

but we cannot be explicit about the second variable that is to

be held fixed, and will simply denote it as x .

From Eq. (9) we then have

pNST

NU

MxMx

,,

(11)

For fixed market values of goods, U be maximized for each

good. Fixed market value means that the goods 1 and 2 are

exchanged in the marketplace subject to the condition [9]

22110 dNpdNp (12)

The maximization of U require that

2

2

1

1

0 dNNUdN

NU

(13)

Combining equation (12) and (13) then yields

tConsNU

ptan

1

(14)



3411

for each good. Thus, as desired, the ratio of value in use to

price is a constant.

Using equation (11) and (14) can be expressed as

tConsNS

pT

NU

p MxMx

tan11

,,

(15)

From Equation (15) the constancy of this ratio for all goods

does not depend on whether the price is included in the

computation of utility. The present formalism helps us focus

on the issue of ‘‘what is held constant. ’’ Let m represent the

value in use (marginal utility per good, at fixed x and M ):

MxNUm

,

(16)

Note that m is specified in monetary units, from equation (14)

and (15), the ratio pm has the same dimensionless value for

all goods.

The ratio pm can be generalized to include the value of

currency, thus permitting the study of saving. Specifically,

define

NxMUm

,

(17)

Then the ratio of value in use to value in exchange for money,

m , takes on the same value as pm for goods.

If we use equation (7) to relate W and U , the analogy to the

development associated with F leads to

pdNdMSdTdW (18)

Where NWp

MW

TWS

,, (19)

It is implicit that two of the three quantities NMT ,, are

held constant in the partial derivatives. From Equation (18)

we may write the functional dependence

NMTWW ,, (20)

Another standard economic relationship states that when an

individual consumer interacts with the market, the price

(marginal cost per good) is determined by the market. We can

obtain this result by assuming that, in equilibrium, the total

wealth of the consumer and of the market is maximized at

fixed temperature and money. Considering the market to be a

reservoir r , we have from Equation (18).

rrrrrrr dNpdMdTSdW (21)

Subject to the condition 0 rdTdT , conservation of

money 0 rdMdM , and conservation of goods

0 rdNdN , we find by adding equation (18) and (21)

that

dNppdMdWdW rrr (22)

The right-hand side of Equation (22) is zero for arbitrary

variations dM and dN only if the value of money to the

consumer is the same as the value of money to the market:

r .

Similarly, for the value of a good, we have rpp . Note that

equation (2) gives the differential

NdpMdpdNdMdW (23)

The consistency of equation (22) and (18) require that

NdpMdSdT 0 (24)

Equation (24) is the analogue of the Gibbs–Duhemrelation.

Among other things, it implies that a decrease in the price of

money or goods (as when the state of economic development

increases) is accompanied by an increase in the economic

temperature. This qualitative behavior is expected from

conventional economic reasoning. Specifically, if all prices

and currency values are increased by a common factor, then

the system does not really change: By Equation (24) the

temperature is simply increased by the common factor.

However, Equation (24) holds for more general variations.

Note that scaling by a factor of 2 means calling a one-dollar

bill a two-dollar bill, etc. To really change the value of the

dollar would require printing more dollars, which is a real cost

that cannot be scaled away.

Equation (24) has an important application. We can write the

change in the Marshallian surplus, TS , as

NdpMdTdSSdTTdSd (25)

We interpret the term TdS as the change in the economic

value of leisure. To see that Ndp is a surplus, note that the

cost of incrementally purchasing goods (where the first goods

are scarce and, hence, costly) is N

pdN0

where Np , the

price of the Nth good, decreases as N increases (that is,

0dNdp . )However, when purchased all at once, the

actual cost to the consumer is Np , the number of goods times

the latest cost per good. The difference is

Np

p

NNdpNppdN

00 (26)

The difference is positive, because for the limits of integration

in Equation (26), dp is negative. Hence Ndp is the

change in the consumer’s surplus of goods.

Equation (25) shows that there are two types of surplus: the

Veblenian surplus of leisure and the Smithian surplus from

efficiency due to specialization. These ideas are present in

economics, but we are unaware of any previous formal

statement that relates the Marshallian, Veblenian, and

Smithian surpluses. Note that the statement of the constancy

of pm for all goods purchased by a given consumer is the

same as the statement that the ratio of Veblenian surplus per

good to the price per good is a constant for all goods

purchased by a given consumer.

The following ‘‘Maxwell relations’’ are an immediate

consequence of the fact that the order of the cross derivatives

of NMSUU ., does not matter. Thus

MSNM NT

Spor

SNU

NSU

,,

22

(27)



3412

NSNM MT

Sor

SMU

MSU

,,

22

(28)

NSVS MNpor

MNU

NMU

,,

22

(29)

In economics, these relations are known as Slutsky conditions

[14, 15]. They guarantee that integrals over dU in

MNS ,, space are path independent. Similar Slutsky

conditions can be derived from W , for which the natural

variables are MT , , and N . Equations (27) and (28) are

new; Equation (29) is already known in a form where the

dependence on entropy is not made explicit. The addition of

the variables T and S helps makes more precise the meaning

of the phrase ‘‘all other quantities held constant. ’’

Boltzmann – Gibbs Distribution

Let us consider a system of many economic agents 1N ,

which may be individuals or corporations. In this paper, we

only consider the case where their number is constant. Each

agent i has some money im and may exchange it with other

agents. It is implied that money is used for some economy

activity, such as buying or selling materials products;

however, we are not interested is that aspects. For us the only

result of interaction between agents i and j is that some

money m changes hands [6]:

mmmmmmmm jijiji ,,, ´´ (30)

Notice that the total amount of money is conserved in each

transaction:

ji mm ´´, ji mm (31)

The local conservation law of money [5] is analogous to the

conservation of energy in collisions between atoms.

We assume that the economic system is closed, i. e. there is no

externalflux of money, thus the total amount of money M in

the system is conserved. Also, in the first part of the paper, we

do not permit any debt, so each agent’s money must be non-

negative: 0im . A similar condition applies to the kinetic

energy of atoms: 0i .

Let us introduce the probability distribution function of

money mP , which is defined so that the number of agents

with money between m and dmm is equal to dmmNP

We are interested in the stationary distribution mPcorresponding to the state of thermodynamic equilibrium. In

this state, an individual agent’s moneymi strongly fluctuates,

but the overall probability distribution mP does not change.

The equilibrium distribution function mP can be derived in

the same manner as the equilibrium distribution function of

energy P in physics [2]. Let us divide the system into two

subsystems 1 and 2. Taking into account that money is

conserved and additive:

21 mmm (32)

Whereas the probability is multiplicative:

21PPP (33)

We conclude that

2121 mPmPmmP (34)

The solution of this equation is

Tm

CemP

(35)

Thus the equilibrium probability distribution of money has the

Boltzmann-Gibbs form. From the normalization conditions

0 01 NMdmMmPanddmmP (36)

We find that TC 1 and NMT

Thus, the effective temperature T is the average amount of

money per agent.

The Boltzmann- Gibbs distribution can be also obtained by

maximizing the entropy of money distribution

mInPmPdmS

0 (37)

Under the constraint of money conservation [2].

Following original Boltzmann´s argument, let us divide the

money axis m0 into small bins of size dm and

number the bins consecutively with the index ,...2,1b .

Let us denote the number of agents in a bin b as bN , the total

number being

1b bNN (38)

The agents in the bin b have money mb , and the total money

is

1b bb NmM (39)

The probability of realization of a certain set of occupation

numbers bN is proportional to the number of ways N

agents can be distributed among the bins preserving the set

bN

This number is

!!

!

21 NNN

(40)

The logarithm of probability is entropy

1!!

b bInNInN (41)

When the number bN are big and Stirling´s formula

NInNInN ! applies, the entropy per agents is

1

1

b bbb bb InPP

NInNNNInN

(42)

Where NN

P bb

is the probability that an agent has money bm .

Using the method of Lagrange multipliers to maximize the

entropy S with respect to the occupation numbers bN with

the constraints on the total money M and the total number of



3413

agents N generates the Boltzmann-Gibbs distribution for

mP [2].

Figure 1: Histogram and points: stationary probability

distribution of money mP . Solid curves: fits to the

Boltzmann- Gibbs law TmmP exp . Vertical lines:

the initial distribution of money.

Theoretical Economic Systems Consider a hypothetical economic system in equilibrium for

which a quantity, say M , is conserved. There is a reasonable

number of arguments [16] which show that certain current

economies can be considered as systems in equilibrium and

some quantities, like the total amount of money in the system,

are conserved during certain periods of time. For the sake of

concreteness we will consider in this work that M is the

conserved money, although our approach can be applied to

any conserved quantity. Suppose that the system is composed

of N agents which compete to acquire a participation m of

M . In real economic systems, the total number of agents is

such a large number that for most applications the limit

N is appropriate. In a closed economic system, the

equilibrium probability distribution (density function) of m is

given by the Boltzmann-Gibbs distribution Tmem ,

whereT is an effective temperature equal to the average

amount of money per agent. The amount of money m that an

agent can earn depends on several additional parameters

l ,..., 21 , which we call microeconomic parameters. Since

the density function can be normalized to 1, we obtain [17]

dexTQxTQ

e TmTm

,,,

(43)

Where xTQ , the partition is function and represents the

set of all microeconomic parameters. Here we have introduced

the notation nxxxx ,..., 21 to denote the possible set of

macroeconomic parameters which can appear after the

integration over the entire domain of definition of the

microeconomic parameters .

Following the standard procedure of statistical

thermodynamics [17], we introduce the concept of mean value

g for any function gg as

dgexTQ

dgg Tm

,

1 (44)

These are the main concepts which are needed in statistical

thermodynamics for the investigation of a system which

depends on the temperature T and macroscopic variables x .

Consider, for instance, the mean value of the money

dmm (45)

And let us compute the total differential md :

dmdmdddmmdmd (46)

The first term of this expression can be further manipulated by

using the definition of the density function (43) in the form

InQInTm (47)

Then, we obtain that 1d and therefore,

0 dd

n

iiidxyTdSmd

1

(48)

Where the entropy S and the ´intensive‘ macroscopic

variables are defined in the standard manner as

dInInS (49)

d

xm

xmy

iii (50)

Clearly, Equation (48) represents the first law of

thermodynamics. Since the definition of temperature and

entropy are in accordance with the concepts of statistical

mechanics, the remaining laws of thermodynamics are also

valid. Similar results can be obtained for any quantity that can

be shown to be conserved in an economic system. This

reflects the fact that different thermodynamic potentials can

be used to describe the same thermodynamic system.

It is useful to calculate explicitly the entropy

dInS (51)

By using the equation (43) in the form

TInQIn (52)

The result can be cast in the form

xTTInQTSmf ,: (53)

So that

ii x

fyTfS

, (54)

This means that the entire information about the

thermodynamic properties of the system is contained in the

expression for the “free money” f which, in turn, is

completely determined by the partition function xTQ , . In

statistical physics this procedure is still used, with excellent



3414

results, to investigate the properties of thermodynamic

systems. We propose to use a similar approach in

econophysics. In fact, to investigate a model for an economic

system one only needs to formulate the explicit dependence of

any conserved quantity, say money m , in terms of the

microeconomic parameters . From m one calculates

the partition xTQ , function and the free money xTf ,

which contains all the thermodynamic in-formation about the

economic system.

In real economic systems, probably very complicated

expressions for m will appear for which analytical

computations are not available. Nevertheless, the calculation

of the above integrals for the partition function can always be

performed by using numerical methods so that the

corresponding thermodynamic properties of the system can be

found qualitatively.

The determination of the quantity m in terms of the

microeconomic parameters is a task that requires the

knowledge of very specific conditions and relationships

within a given economic system. The first step consists in

identifying the microeconomic parameters which

influences the capacity of an individual agent to compete for a

share m of the conserved quantity M . Then, it is necessary to

establish how this influence should be represented

mathematically so that m becomes a well-defined function of

.

Another aspect that must be considered is the fact that in a

realistic economic system not all agents are equivalent. For

instance, an agent represented by an individual who works at a

factory for a fixed yearly income would be considerably

different from an agent represented by the firm to which the

factory belongs. An important result of econophysics is that

the Botzmann-Gibbs distribution is not affected by the

specific characteristics of the agents involved in the economic

model [18]. For the statistical thermodynamic approach we

are proposing in this work this means that we can decompose

the quantity m into classes ..., III mmm and different

classes can be described by different functions of different

sets of microeconomic parameters. The formalism of

statistical thermodynamics allows us to consider, in principle,

all possible economic configurations as far as m is a well-

defined function of .

In the following subsections we will study several

hypothetical economic systems in which m is given as simple

ordinary functions of the microeconomic parameters. We

expect, however, that these simple examples will find some

applications in the context of economic systems with

sufficiently well-defined microeconomic parameters.

Although the function m can represent any conserved

economic quantity, for the sake of concreteness, we will m

assume that it represents the money and from now will be

referred as to the money function.

The simplest model corresponds to the case

tConscm tan0

Then, the partition function (43) is given by

,...,, 21

0

nT

ceTQ

(55)

Where nid ii ,...,2,1, , represents the

macroeconomic parameters.

The calculation of the thermodynamics variables, according to

Equation (54), yields

iii

TfyInSTIncf

,,0 (56)

Furthermore, from the definition of free money f we

conclude that ,0cm i.e., the mean value of money is a

constant, as expected. This economic model is considerably

simple. Each agent possesses the same amount of money 0c ,

the entropy does not depend on the mean value of the money

0c , and the state equations are ,0cm and Ty ii .

The system is completely homogeneous in the sense that each

agent starts with a given amount of money, 0c , and ends up

with the same amount. Probably, the only way to simulate

such an economic system would be by demanding that agents

do not interchange money; this is not a very realistic situation.

Indeed, the fact that the entropy is a constant, that does not

depend on the mean value of the money, allows us to

renormalize the macroscopic parameters in such a way that

1ii d (57)

for each i , so that the total entropy vanishes.

In this case, from Equation (55) we see that 0cf and the

corresponding equations of state are compatible with the limit

0T . This resembles the argumentation used in the

description of the third law of thermodynamics. This

observation indicates that a completely homogeneous

economic system is not realizable as a consequence of the

third law of thermodynamics.

Consider now the function

11cm (58)

Where 1c is a positive constant. The corresponding partition

function can be written as

0 11

1

1

1111

,cTdedeTQ T

cT

c

(59)

The relevant thermodynamics variables follow equation (53)

and (54)

1111

1,

cTInS

cTTInf (60)

,1,,01

jTyyj

j (61)

And the conservation law (48) becomes

n

j j

jdTTdSmd

2

(62)



3415

Moreover, comparing the above results with the definition of

f , it can be shown that Tm , and so the fundamentals

thermodynamics equation in the entropic representation [19]

can be written as

n

jjIn

cm

InS21

1 (63)

This expression relates all the extensive variables of the

system and it can be used to derive all the equations of state in

a manner equivalent to that given in Equation (61). Notice that

in this case the entropy is proportional to the temperature

(mean value of money) so that an increase of the average

money per agent is necessarily associated within increase of

entropy. This observation is in agreement with the second law

of thermodynamics. Notice, furthermore, that in a limiting

case, similar to that considered in the first example given

above, it is possible to renormalize the macroeconomic

parameters j such that the last term of the fundamental

equation (63) vanishes. Nevertheless, in order to reach the

minimum value of the entropy it is necessary to consider the

limit 0T . Again, we consider this result as an indication

of the validity of the third law of thermodynamics.

In Equation (59) we have chosen the interval ,0 for the

integration along the variable 1 . As a consequence the

macroscopic variable 1 vanishes from the final expression

for the fundamental equation (62), and consequently 01 y .

However, it is also possible to consider the interval

max

1

min

1 , so that the macroscopic variables min

1 and

max

1 reappear in the fundamental equation and can be used

as extensive variables which enter the conservation law (48).

In a realistic economic system the interval of integration will

depend on the economic significance of the microeconomic

parameter 1 . For the sake of simplicity, we choose in this

work the former case in which the corresponding

macroeconomic parameter does not enter the analysis.

It is interesting to analysis the most general linear system for

which

n

iiiccm

1

0 (64)

Where 1,0 cc , … are positive real constants. It is then straight

forward to calculate thermodynamics variables

,...,1

21

0

nnT

cccccTe

cTQ

(65)

And the relevant thermodynamics variables

,,0 cTInnS

cTTIncf

nn

(66)

.,0 0 nTcmyi (67)

All the macroscopic parameters vanish and the system

depends only on the temperature. However, the total number

of macroscopic parameters n does enter the expression for

the entropy so that to increase the mean value of the money by

the amount

12 TTnm , (68)

it is necessary to increase entropy by an amount

1

2

TTnInS (69)

Both amounts are proportional to the total number of

macroscopic parameters. Another consequence of this analysis

is that once the constants 0c and n are fixed, it is not possible

to change the mean value of the money without changing the

temperature of the system. This means that an isothermal

positive change of m is possible only by increasing the

total amount of money in the system.

Next, consider the quadratic function 2

11cm which

generates the partition function

21

1

21

1

, Tc

TQ

(70)

Where we have considered the microeconomics parameter 1

in the interval , . The corresponding thermodynamics

variables are

,2

,,0,12

11

11

TmTyyIncTInS

jj

(71)

Which can be put together in the fundamental equation

.2

12

1

11

In

cm

InS

(72)

Again we see that the effect of considering the extreme values

of the parameter 1 is that the corresponding macroscopic

variable 1 does not enter the expressions for the

thermodynamic variables and, consequently, the

corresponding intensive thermodynamic variable vanishes.

Furthermore, it is evident that the power of 1 in the money

function leads to a decrease of the mean value m .

To investigate the general case, we consider the monomial

functional dependence

11cm (73)

with being an arbitrary real constant. A straightforward

calculation leads to the following partition function and

thermodynamic variables:

1,

1

1cTTQ (74)

,,1

11

11

TmIncTInS

(75)

and the intensive variables iy are given as in the previous

case. It follows that 0 in order for the mean value of the



3416

money to be positive. If 1 , the mean value of the money

decreases, whereas it increases for 10 . In such a

hypothetical system, a way to increase the amount of money

per agent would be to identify the microeconomic parameter

1 and apply the measures which are necessary for the money

function m to become 1m with 1 .

If we consider a transition of an economic system from a state

characterized by the parameter 1 to a new state with

parameter 2 , maintaining the same temperature, the mean

value of the money undergoes a change

Tm

12

11

(76)

so that for a positive change we must require that 12 .

Moreover, if we desire a positive change by an amount greater

than the average money per agent Tm , we must

demand that

12 11

Even if the initial state corresponds to a linear system

11 , in which no increase of m is possible, we can

reach a state of greater mean m value by demanding that

212 . Of course, for a positive change of m the "price”

to be paid will result in an increase of entropy by an amount

which is proportional to the coefficient 12 11

Thermodynamics Monetary System Now we might imagine that the value of economic output

flow rate MG in monetary terms will be equated to its price

multiplied by a volume rate of flow zV

ZMM VPG (77)

To reconcile economic concepts with thermo-economic ones

we imagine that at a particular point in time in production the

two output value flowrates are deemed to be equivalent to

each other, though of course they are defined in very different

ways:

0GGM (78)

0G is defined in terms of some measure or complex of

measures of final productive content.

For money system, the ideal economic equation is expressed

as in above equation (78)

MMMZM TkNVP (79)

Where MP is the price level, ZV is output volume, MN is

the number of currency instruments in circulation, Mk is a

nominal monetary standard (£1, €1, $1, etc.) and MT is an

Index of Trading Value, here the velocity of circulation of

money.

This equation is a re-statement of the general quantity theory

of money. The quantity theory is commonly written as

MVPY (80) where P is price level in an economic system, Y is output in

volume terms, M is the quantity of money in circulation and

V is the velocity of circulation. While the left-hand sides of

the equations are comparable, the right-hand side requires

additional clarification. To obtain an exact comparison with

the quantity theory, equation (79) can be written as:

TNkPV (81)

Where Nk is equivalent to money M in circulation and the

index of trading value T in a thermodynamic monetary

system is equivalent to the velocity of circulation V in a

traditional monetary system. The equation has a very similar

format to the ideal gas equation. Pikler[20] has highlighted the

connections between the velocity of circulation and

temperature.

Thus, in equation (79), there are four variables to consider,

and in differential form equation (79), from now on without

the subscripts, can be written as:

TdT

NdN

VdV

PdP

(82)

The nominal currency value k , is fixed by definition, though

it can change its effective value through inflation and

international comparison.

Before developing a thermodynamic representation of a

money system, appoint should be made concerning the

number of variables. In a non-flow thermodynamic system,

both the constant k and the number of molecules N are

fixed, and there are therefore only three variables left to

consider, pressure P , volume V and temperatureT . In a

thermodynamic flow system, on the other hand, there are four

variables left (other than the constant k ):molecular flow N ,

volume flowV , pressure P and temperatureT , but these are

effectively reduced to three, by replacing volume flow V and

molecular flow N with specific volume NVv . Thus

again only three variables remain, enabling thermodynamic

analysis to proceed.

To develop a thermodynamic analysis of a money system, a

similar requirement is necessary; that is, to reduce the four

factors in equation (82) plus k (making five) to three,

preferably without losing output volume flowV .

The probability is that any change in money instrument stock

N may find its way into changes in all three of the other

variables, price, output volume or velocity of circulation,

depending upon the relative elasticity between the three and

with money, and the degree to which a money system is out of

kilter with the stable state, such as the existence of excess

money or high inflation.

Therefore, the preferred arrangement to develop the

thermodynamic characteristics, although not perfect, is to

transpose money units N to the left-hand side of the

equation, and divide output price level P by the number N

of monetary units in circulation to give a Specific Price



3417

NPPN . Presentations of the inverse PNNP are

also given at each stage however for completeness, as the

results are the same.

The equation (79) could be written as:

kTVPN (83)

Or in the alternative:

kTNV P (84)

By differentiating equation (83) and dividing by kTVPN

we have:

TdT

VdV

PdP

N

N (85)

Where

N

N

PdP

also equals

P

P

NdN

.

Figure 2: Velocity of circulation, specific price, specific

money stock and output volume

Figure 2 illustrates the relationships between the factors. A

change in velocity of circulation T can arise either from a

change in specific price NP or a change in output volumeV ,

or both. And no change in the velocity of circulation T will

occur if a change in specific price NP is balanced by an

equivalent change in output volume V .

Figure 3: Price – Volume Relationships

As illustrated at figure 3, a number of relationships between

price and volume are current in a thermodynamic

presentation.

Figure 4: Specific Price –Output Volume Relationships

Figure 4 illustrates the same relationships, but in terms of

specific price NP .

For an economy where no change in output volume occurs (

V = Constant), increases/decreases in specific price NP are

matched by an equivalent change in the velocity of circulation

T . Such changes in specific price NP involve a change in the

relationship between actual price level P and the money

stock N .

At the other extreme, where no change in the specific price

NP occurs, changes in real output volume V are matched by

appropriate changes in the velocity of circulationT . Should a

movement in money supply N occur, this will be balanced

by a change in price P .

If the velocity of circulation T remains constant, then any

change in output volume V will result in an offsetting change

in the specific price NP , accompanied by a change in the

relationship of price P to money supply N .



3418

Last, a further relationship between the variables can exist,

which in this book we have called the Polytropic case

tConsVP nN tan ), involving an Elastic Index n , where

changes in all of the factors can take place, but in a complex

manner. This is shown at figure 4.

The polytrophic equation tConsVP nN tan can be

adapted to meet all of the possible processes. A constant

specific price process

0

N

N

PdP

for example is a

polytrophic process with the elastic index n set at zero, and a

constant volume process is one with the elastic index set at

. The iso trading model tConsVPN tan has an

index of 1, with no change in velocity of circulation T .

Because the monetary stock system is effectively one stock

system, though serving all the other sectors, we could use the

polytrophic case to describe its dynamics, since this covers all

the other price-volume relationships.

By combining equation (85) with the polytrophic equation

CVP nN , the following equations describe the polytrophic

case:

VdVn

PdP

VV

PP

N

Nn

N

N

1

2

1

2

(86)

N

Nn

n

N

N

PdP

nn

TdT

PP

TT

11

1

2

1

2

(87)

VdVn

TdT

VV

TT

n

1

1

1

2

1

2

(88)

Thus far, we have not specified how the relationships impact

on the way in which an economy moves and what drives the

relationships, only that changes in one or more of the factors

will be reflected by changes to the others to enable the

equation of state set out at equations (79), (82) and(85) to

balance out.

The above regressions for the two economies of course

average out short-term variations.

Because price P has been divided by money stock N to

compute a specific price NP , and vice-versa for specific

money PN , the relative elasticities between the variables in

the polytrophic case will be different compared to the position

with these variables being separate.

By utilizing the differential form of equation (86) it is possible

to calculate the short-term quarter-on-quarter variation in the

elastic index n for the two economies.

VdVn

NdNand

VdVn

PdP

P

P

N

N

The three charts at figure 5 illustrate the effect of a change in

the elastic index n between specific price NP , output volume

V and velocity of circulationT .

To finish this section, we refer back to the polytrophic

equation (86). Splitting this into its component parts, we have:

tConsVNPVP nn

N tan

(89)

And taking logs and differentiating we have:

VdVn

NdN

PdP

(90)

Figure 5: Elasticity between specific Price NP , Output

Volume V and Velocity of CirculationT .



3419

Conclusion The paper highlights the analogues of the thermodynamics

axioms to money dynamics. Within this context the idea of a

money system is developed, based on the origin from

thermodynamics. In this work we propose to apply the

standard methods of statistical thermodynamics in order to

investigate the structure and behavior of economic systems.

Here we presented to cover the Boltzmann – Gibbs

distribution of money in thermodynamics of a big ensemble of

economic agents realistic deterministic strategies with money

conservation. From this paper, we introduced concepts

borrowed from statistical thermodynamics, like entropy into

money. The outcomes of this research analyzed started with

and we approached through the analogy relation between

statistical thermodynamics and economics system then we

demonstrated how the Boltzmann – Gibbs distribution

emerges in economics models. Followed by we extended this

approach through the hypothetical economic systems in a

linear and nonlinear system. Finally we constructed the

thermodynamics monetary system at Polytrophic Constant

and this paper ended with conclusion.

References

[1] J. D. Farmer, Computing in Science and Engineering

1, 26 (1999); R. N. Mantegna, H. E. Stanley, An

Introduction to Econophysics (Cambridge University

Press, Cambridge, (2000).

[2] G. H. Wannier, Statistical Physics (Dover, New

York, (1987).

[3] C. Tsallis, J. Stat. Phys. 52, 479 (1988); Braz. J.

Phys. 29, 1 (1999).

[4] D. M. Mueth, H. M. Jaeger, S. R. Nagel, Phys. Rev.

E 57, 3164 (1998); M. L. Nguyen, S. N.

Coppersmith, Phys. Rev. 59, 5870 (1999).

[5] M. Shubik, in The Economy as an Evolving

Complex System II, edited by W. B. Arthur, S. N.

Durlauf, D. A. Lane (Addison-Wesley, Reading,

1997), p. 263.

[6] S. Ispolatov, P. L. Krapivsky, and S. Redner, Wealth

distribution in asset exchange models, Eur. Phys. J. B

2 (1998) 267.

[7] C. R. McConnell, S. L. Brue, Economics: Principles,

Problems, and Policies (McGraw-Hill, New York,

(1996).

[8] E. W. Montroll, M. F. Shlesinger, Proc. Natl. Acad.

Sci. USA 79, 3380 (1982); O. Malcai, O. Biham, S.

Solomon, Phys. Rev. E 60, 1299 (1999); D. Sornette,

R. Cont, J. Phys. I France 7, 431 (1997); J. -P.

Bouchaud, M. Mezard, cond-mat/0002374.

[9] P. Samuelson, Economics (McGraw–Hill, New

York, 1970), 8th ed., p. 410.

[10] E. Silberberg, The Structure of Economics: A

Mathematical Analysis (McGraw–Hill, New York,

1990), p. 396.

[11] E. K. Browning and J. M. Browning, Microeconomic

Theory and Applications (HarperCollins, New York,

1992), 4th ed., p. 87.

[12] W. Benton, Encyclopedia Brittanica(Chicago, 1966),

Vol. 22, p. 92.

[13] Adam Smith, An Inquiry into the Nature and Causes

of the Wealth of Nations, published in Dublin,

printed for Messrs. Whitestone and others (1776).

[14] W. E. Diewert, ‘‘Generalized Slutsky conditions for

aggregate consumer demand functions, ’’ J. Econ.

Theory 15, 353–362 (1977).

[15] A. Barten, ‘‘Evidence on the Slutsky conditions for

demand equations, ’’ Rev. Econ. Statistics 49, 77–84

(1967).

[16] V. M. Yakovenko, Econophysics, Statistical

Mechanics Approach to, (2007) arXiv:q-fin.

ST/0709. 3662.

[17] K. Huang, Statistical mechanics, John Wiley & Sons,

New York, 1987.

[18] A. Dragulescu and V. M. Yakovenko, Statistical

mechanics of money, Eur. Phys. J. B 17 (2000) 723.

[19] H. B. Callen, Thermodynamics and an Introduction

to Thermostatics (Wiley, New York, 1985).

[20] Pikler, A. G. (1954) 'Optimum allocation in

econometrics and physics',

WelwirtschaftlichesArchiv.

[21] E. M. Lifshitz, L. P. Pitaevski, Physical Kinetics

(Pergamon Press, New York, (1993).

[22] A. Chakraborti and B. K. Chakrabarti, Statistical

mechanics of money: how saving propensity affects

its distribution, Eur. Phys. J. B 17 (2000) 167.

[23] M. Aoki, New Approaches to Macroeconomic

Modeling (Cambridge University Press, Cambridge,

(1996).

[24] V. Pareto, Cours d' EconomiePolitique (Lausanne

and Paris, 1897); G. J. Stigler, J. Business 37, 117

(1964).

[25] S. Moss de Oliveira, P. M. C. de Oliveira, D.

Stauffer, Evolution, Money, War and Computers,

edited by B. G. Teubner (Stuttgart, 1999); The

Economy as an Evolving Complex system, edited by

P. W. Anderson, K. J. Arrow, D. Pines (Addison-

Wesley, Redwood City, (1988).

[26] M. Levy, S. Solomon, Physica A 242, 90 (1997).

[27] M. Plischke, B. Bergersen, Equilibrium Statistical

Physics, 2nd edn. (World Scientific, Singapore,

1994); S. K. Ma, Statistical Mechanics (World

Scientific, Singapore, (1985).

[28] J. M. Keynes, The General Theory of Employment,

Interest and Money (The Royal Economic Society,

Macmillan Press, London, (1973).

[29] G. Guillaume and E. Guillaume, Sur le fundements

deleconomiquerationelle, Gautier-Villars, Paris,

1932.

[30] E. Samanidou, E. Zschischang, D. Stauffer, and T.

Lux, Agent-based models of financial markets, Rep.

Prog. Phys. 70 (2007) 409.

[31] 5. W. M. Saslow, An economic analogy to

thermodynamics, Am. J. Phys. 67 (1999) 1239.

[32] H. E. Stanley, Anomalous fluctuations in the

dynamics of complex systems: from DNA and

physiology to econophysics, PhysicaA 224 (1996)

302.



3420

[33] J. P. Bouchaud and M. M´ezard, Wealth

condensation in a simple model of economy,

PhysicaA 282 (2000) 536.

[34] A. C. Silva and V. M. Yakovenko, Temporal

evolution of the ‘thermal’ and ‘superthermal’ income

classes in the USA during 1983-2001, Europhys.

Lett. 69 (2005) 304.

[35] A. Banerjee, V. M. Yakovenko, and T. Di Matteo, A

study of the personal income distribution in

Australia, PhysicaA 370 (2006) 54.

[36] H. Quevedo, Geometrothermodynamics, J. Math.

Phys. 48 (2007) 013506.

statistical thermodynamics of money (thermoney)statistical thermodynamics, like entropy into money....

Documents