Elementary kinetic exchange models

In the second half of the XIXth century physicists, of whom probably the most well known are Maxwell and Boltzman, worked on the explanation of empirically discovered laws of thermodynamics. While working on this problem they developed a simple model to reproduce the collisions of particles in the ideal gasses. This simple model allowed to analytically derive the distribution of energy and velocities in gasses and to lay foundations for the statistical physics. In the context of Physics of Risk it is worthwhile to mention that Maxwell and Boltzman relied not only on the empirical works by other physicists, but also on the demographic data! Boltzmann even wrote that "molecules are like so many individuals, having the most various states of motion" [1, 2]. Inspired by this quote we will briefly review, while relying on [2], some of the simplest kinetic models and their applications to modeling of socio-economic systems.

Elementary kinetic model

One of the advantages of the kinetic models is that they are easily implemented and modified. Actually in most cases the basic scheme behind the kinetic models is the same - randomly moving particles interact and exchange their kinetic energy. The only freedom of choice here is the mechanism of the exchange.

Let us assume that we have a fixed number of particles, \( N \), in the system. Also let us assume that the particles have certain kinetic energy, \( w_i(t) \), at any given time \( t \). Because the particles interact only during collisions and due to the fact that there are no external forces acting on the system, the internal energy of the system should remain the same and be equal to the sum of kinetic energies of the particles, \( U = \sum_{i=1}^N w_i(t) \).

At each discrete time step (\( t=0,1,2,\ldots \)) let us select two random particles, given indexes \( i \) and \( j \), and assume that only these two particles are interacting at this moment. Evidently the kinetic energies after the interaction will depend on the energies particles had prior to the interaction:

\begin{equation} w_i(t+1) = w_i(t) - \Delta w_{ij} , \quad w_j(t+1) =w_j(t) + \Delta w_{ij} . \label{wchange} \end{equation}

\( \Delta w_{ij} \), in general case, may have many distinct forms. Yet we can talk about some of them in the physical context. E.g., if particles move only in single dimension (balls on a stiff metal rod), then all of the collisions will be head-on, thus leading to \( \Delta w_{ij} = w_i(t) - w_j(t) \). Yet the one dimensional case is not interesting and completely trivial as the distribution of absolute speeds and energies will not change. More interesting cases are those of higher dimensions, because in such case head-on collisions become rare and larger variety of collisions become probable. Due to the molecular chaos hypothesis for a two dimension case we can assume that [2]:

\begin{equation} \Delta w_{ij} = (1-\varepsilon) w_i(t) - \varepsilon w_j(t) , \label{elementary} \end{equation}

were \( \varepsilon \) is a random variable uniformly distributed in \( [0,1] \). Note the general similarity of the two and one dimensional exchange equations. Three and higher dimensional cases should be more or less similar [2], but one should take into account that the probability of head-on collisions decreases proportional to the dimensionality of the system, \( p_c\sim 1/d \) (here \( d \) stands for the dimension).

Note that by putting \eqref{elementary} into \eqref{wchange} one obtains

\begin{equation} w_i(t+1) = \varepsilon [ w_i(t) + w_j(t) ] , \quad w_j(t+1) = (1-\varepsilon) [ w_i(t) + w_j(t) ] . \end{equation}

As we can see in the above, in this case, after collision, we have total kinetic energy of both particles being uniformly distributed among them. Thus in the literature this model is known as random reshuffling or reshuffling model [2].

In an HTML5 applet below you can convince yourself that the stationary distribution of this model is Boltzmann-Gibbs distribution (given by red curve).

Constant exchange model

Boltzmann-Gibbs distribution can be also easily obtained if the exchange rate is constant, namely if:

\begin{equation} \Delta w_{ij} = w_0 , \end{equation}

here \( w_0 \) is a model parameter describing the amount of energy exchange during each interaction.

Fixed saving and borrowing

There are at least two more simple modifications, which might be implemented in the model without making it overly complex. Interestingly enough these two modifications are the exact opposites of each other. Our first option is to allow the particles to borrow energy up to \( w_{max} \) (it is the same as to allow the negative energies). The second options is to force particles to conserve certain amount of energy, \( w_{min} \). The previous definition of \( \Delta w_{ij} \), \eqref{elementary}, may be used, one just needs to alter the \( w_i(t) \) and \( w_j(t) \) values prior to the actual exchange (to increase them by \( w_{max} \) or by decreasing them by \( w_{min} \)).

In the HTML5 applet below we have single parameter \( w_{mm} \), which stands for both \( w_{max} \) and \( w_{min} \). If \( w_{mm}>0 \), then the fixed saving model is used (with \( w_{min} = w_{mm} \)). In the opposite case the borrowing model is used (with \( w_{max} = -w_{mm} \)). Note that if \( w_{max} \rightarrow \infty \), then the stationary distribution approaches uniform distribution and the system becomes less stable. While if \( w_{min} \rightarrow 1 \) (approaching initial energy of the particles), the system remains in the initial state.

We would like to draw your attention to the fact that these kind of borrowing and saving rules are somewhat simplistic and are not very realistic in context of modeling real socio-economic systems. Though even from this simple model one can draw a correct conclusion - limitless borrowing may destabilize the system. In the nearest future we promise to comeback with at least a more sophisticated saving rules.

"All in" exchange

Another possible variation of the model is to implement the "All in" exchange. Namely the particle with less energy "gambles" by "betting" its whole energy (same as "all in" in poker or similar betting game):

\begin{equation} \Delta w_{ij} = \mathop{min} (w_i, w_j) . \end{equation}

It should be evident that this model is quite different, because sooner or later all energy will be accumulated by a single particle, while all others will have no energy. See HTML5 applet below.

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.