Analytical treatment of kinetic exchange models

Kinetic exchange models are very simple and powerful tool to understand the processes in the ideal gasses. These simple models enabled Boltzmann to formulate the principles of statistical physics [1, 2]. In previous text we already talked about some of the simplest models, but we did not write about a very important topic - their analytical treatment. It appears that analytical treatment is one of the most serious drawbacks of these models - in certain cases the equations become very complex or even analytically unsolvable. While, our usual approach, one-step processes (previously discussed on Physics of Risk from the point of view of the Kirman model) can be treated analytically with ease in most of the cases [3, 4]. In this text we will discuss two main techniques used to analytically obtain statistical features of the kinetic exchange models.

Master equation

As is common in statistical physics we can use the Master equation to describe the interaction between two particles. Let us assume that before interactions the particles have energies \( (u_i, u_j) \) and after the interaction \( (u_i', u_j') \), In such case:

\begin{equation} \partial_t f(u) = \left[ \partial_t f(u_i) + \partial_t f(u_j) + \partial_t f(u_i') + \partial_t f(u_j')\right]_{u_i,u_j,u_i',u_j' = u} . \end{equation}

It should be straightforward to see that this Master equation includes terms of two types - first type is related to the situation before interaction, while the other describes the changes after interaction. Terms "before interaction" should describe decrease in probability density, as the particles may be selected from the vicinity of \( u \), while the terms after "the interaction" should increase the probability density, as the particles may arrive to the vicinity of \( u \). As the particles are selected uniformly at random, then the probability density will decrease proportionally to the probability to find the particle in the vicinity (namely to itself):

\begin{equation} \partial_t f(u_i) \sim - f(u_i) , \quad \partial_tf(u_j) \sim - f(u_j) . \end{equation}

We also know that after the interaction the energy of the particles will be uniformly distributed in the interval \( [ 0 , u_i + u_j] \), thus the probability that particle will arrive to a certain vicinity is:

\begin{equation} \partial_t f(u_i') \sim \int_{u_i'}^\infty \left[\int_0^U f(u_i) f(U-u_i) \mathrm{d} u_i \right]\frac{\mathrm{d} U}{U}, \end{equation}

\begin{equation} \partial_t f(u_j') \sim \int_{u_j'}^\infty \left[\int_0^U f(u_i) f(U-u_i) \mathrm{d} u_i \right]\frac{\mathrm{d} U}{U} . \end{equation}

In the above the external integral means that the final distribution of energies of the particles is uniform. While the internal integral describes the probability to find two particles with a total energy of \( U= u_i +u_j \).

Thus by putting these into the Master equation we obtain:

\begin{equation} N \partial_t f(u) = - 2 f(u) + 2 \int_{u}^\infty \left[\int_0^U f(u_i) f(U-u_i) \mathrm{d} u_i \right]\frac{\mathrm{d} U}{U} . \end{equation}

It should be pretty straightforward to see that \( f(u) = \exp(-u) \) is a stationary probability density function. The problem is that this case is very simple, while in a more complex cases the integrals may be extremely hard to evaluate both analytically and numerically.

Obtaining the moments of the stationary distribution

In a more complex cases we may attempt to find the distinct moments of the stationary distribution [5]. E. q., in case of kinetic exchange model with savings we can write down:

\begin{equation} \langle X^m \rangle = \langle [ \kappa X_1 + \varepsilon(1-\kappa ) (X_1 + X_2) ]^m \rangle . \end{equation}

After the expansion and simplification we obtain:

\begin{equation} \langle X^m \rangle = \sum_{k=0}^m \binom{m}{k}\frac{\kappa ^{m-k} (1-\kappa )^k}{k+1} \sum_{p=0}^{k}\binom{k}{p} \langle X^{m-p} \rangle \langle X^p \rangle . \end{equation}

From this equation we can work out the moments one by one. Let us recall that we have previously seen that the Gamma distribution is a good approximation of the stationary probability density of this model. Let us check! As you can see up until the third moment everything seems fine:

\begin{equation} \langle X^2 \rangle = \frac{2+\kappa }{1+2 \kappa } =\langle x^2_\gamma \rangle , \quad \langle X^3 \rangle =\frac{3 (2+\kappa )}{(1 + 2 \kappa ^2)} = \langle x^3_\gamma\rangle , \end{equation}

while the fourth moments are not equal, though the difference is negligible:

\begin{equation} \langle X^4 \rangle \neq \langle x^4_\gamma \rangle ,\quad \left| 1 - \frac{\langle x^4_\gamma\rangle}{\langle X^4\rangle} \right| < 10^{-3}, \: \forall \kappa \in [0,1] . \end{equation}

Thus as we can see the stationary distribution is clearly not a Gamma distribution, though it is rather close approximation. Sadly it is impossible to guess the form of the actual stationary distribution.

References

• P. Ball. The physical modelling of society: A historical perspective. Physica A 314: 1-14 (2002).
• M. Patriarca, A. Chakraborti. Kinetic exchange models: From molecular physics to social science. American Journal of Physics 81: 618-623 (2013). doi: 10.1119/1.4807852.
• M. Aoki, H. Yoshikawa. Reconstructing Macroeconomics: A Perspektive from Statistical Physics and Combinatorial Stochastic Processes. Cambridge University Press, 2007.
• N. G. van Kampen. Stochastic process in Physics and Chemistry. North Holland, Amsterdam, 2007.
• M. Lallouache. Kinetic wealth exchange models: some analytical aspects. Ecole Centrale Paris, Master thesis, 2010.