Asymptotic behavior of a stochastic particle system
of 5 neighbors

Discrete particle systems have been studied for various fields such as physics, engineering, and mathematics. Especially, dependency of the momentum of particles on their density in the asymptotic behavior have been the one of important topics for analyzing particle systems[1, 2]. For example, asymmetric simple exclusion process (ASEP) is a popular discrete stochastic particle system which is a multibody random walk model where each particle moves stochastically along one-dimensional lattice space[3, 4, 5]. Its steady state is discussed relating to orthogonal polynomials[6]. Nagel-Schreckenberg model is another interesting example related to traffic flow[7, 8]. Nagel and Schreckenberg theoretically analyzed mechanism of traffic jam, which depends on occupancy of cars in a traffic way.

In this article, we investigate the asymptotic dynamics of the stochastic system and discuss the dependency of mean flux on other quantities. Mean flux of ASEP is determined by particle density uniquely. However, that of our system depends not only particle density but also another conserved quantity.

First, we introduce a deterministic discrete particle system described by

$$u_{j}^{n+1}=u_{j}^{n}+q\left(u_{j-2}^{n},u_{j-1}^{n},u_{j}^{n},u_{j+1}^{n}\right)-q\left(u_{j-1}^{n},u_{j}^{n},u_{j+1}^{n},u_{j+2}^{n}\right).$$

(1)

The variable $\displaystyle u$ is binary which takes value $\displaystyle 0$ or $\displaystyle 1$, $\displaystyle j$ is an integer site number and $\displaystyle n$ is an integer time. The function $\displaystyle q$ is a flux written by Table 1.

We assume the periodic boundary condition for space sites with a period $\displaystyle L$. From the above evolution equation, it is easily shown that

$$\sum_{j=1}^{L}u_{j}^{n+1}=\sum_{j=1}^{L}u_{j}^{n},\ \ \ \sum_{j=1}^{L}{u_{j-1}^{n+1}u_{j}^{n+1}\left(1-u_{j+1}^{n+1}\right)}=\sum_{j=1}^{L}{u_{j-1}^{n}u_{j}^{n}\left(1-u_{j+1}^{n}\right)}.$$

(2)

We use a notation $\displaystyle m_{x_{1}x_{2}\cdots x_{k}}$ which expresses the number of local patterns of $\displaystyle x_{1}x_{2}\cdots x_{k}$ over $\displaystyle L$ sites. The notation $\displaystyle\rho$ is a density of those patterns ($\displaystyle\rho_{x_{1}x_{2}\cdots x_{k}}=m_{x_{1}x_{2}\cdots x_{k}}/L$). Thus, above laws in (2) show that $\displaystyle\rho_{1}$ and $\displaystyle\rho_{110}$ are conserved quantities. Mean flux over the space sites of the asymptotic behavior is defined by

$$Q=\lim_{n\to\infty}{\frac{1}{L}\sum_{j=1}^{L}q\left(u_{j-2}^{n},u_{j-1}^{n},u_{j}^{n},u_{j+1}^{n}\right)}.$$

(3)

Table 1 means the following motion rule of particles.

• •

  An isolated particle (010) does not move.

• •

  For a pair of adjacent two particles (0110), both particles move.

• •

  For a sequence of more than two particles ($\displaystyle 011\ldots 10$), particles other than the leftmost move.

Figure 1 shows an example of solution to this system.

The mean flux Q, which is a mean momentum of particles, depends uniquely on a pair of the densities $\displaystyle\rho_{1}$ and $\displaystyle\rho_{110}$ as follows[9]:

$$Q=\max{\left(2\rho_{1}-1,2\rho_{110}\right)}.$$

(4)

Figure 2 shows the three-dimensional ‘fundamental diagram’ obtained by (4). The domain $\displaystyle(\rho_{1},\rho_{110})$ is restricted by $\displaystyle 2\rho_{110}\leq\rho_{1}\leq 1-\rho_{110}$ considering the relation between $\displaystyle m_{1}$ and $\displaystyle m_{110}$. The usual fundamental diagram is defined by the relation between mean flux and density. Since mean flux of this system depends on two independent quantities ($\displaystyle\rho_{1}$ and $\displaystyle\rho_{110}$), the diagram becomes three-dimensional.

Second, we introduce a stochastic parameter to the above deterministic system and propose a conjecture about asymptotic distribution of the stochastic system. Based on the conjecture, expected values of the mean flux can be derived. Moreover, we confirm that the theoretical formula (4) in the deterministic case can be derived by the limit of the stochastic parameter.

Contents of this article are as follows: In the section 2, we propose stochastic extension of the particle system (1) and derive an explicit formula of mean flux of the asymptotic behavior. In the section 3 we introduce a conjecture about asymptotic distribution of the stochastic system which is derived by eigenvalue problems of transition matrices[10]. Finally, we take the limit of stochastic parameter to obtain the deterministic profile of the fundamental diagram shown in Figure 1.

We introduce an external variable $\displaystyle a$ into Table 1 preserving the conservation laws of (2) for $\displaystyle\rho_{1}$ and $\displaystyle\rho_{110}$, and obtain a stochastic particle system given by equation (5) and Table 2.

$$u_{j}^{n+1}=u_{j}^{n}+q\left(u_{j-2}^{n},u_{j-1}^{n},u_{j}^{n},u_{j+1}^{n}\right)-q\left(u_{j-1}^{n},u_{j}^{n},u_{j+1}^{n},u_{j+2}^{n}\right).$$

(5)

The variable $\displaystyle a$ is a stochastic parameter defined by

$$a=\begin{cases}0&\text{(with probability $\displaystyle\alpha$)}\\ 1&\text{(with probability $\displaystyle 1-\alpha$).}\end{cases}$$

(6)

Figure 3 shows an example of time evolution, and Figure 4 shows numerical results of mean flux of the stochastic system. Comparing with Figure 2, surface of mean flux of the stochastic system converges to that of the deterministic system (1) along $\displaystyle\alpha\to 1$. Figure 5 shows a sectional graph of $\displaystyle\rho_{110}=7/60$ and $\displaystyle\alpha=0.5,\ 0.7,\ 0.9,\ 1$.

In order to analyze this stochastic system, we express the time evolution using a new variable $\displaystyle v_{j}^{n}$ given by $\displaystyle v_{j}^{n}=u_{n-j}^{n}$. The equation is

$$v_{j}^{n+1}=v_{j}^{n}+q\left(v_{j-3}^{n},v_{j-2}^{n},v_{j-1}^{n},v_{j}^{n}\right)-q\left(v_{j-2}^{n},v_{j-1}^{n},v_{j}^{n},v_{j+1}^{n}\right),$$

(7)

and $\displaystyle q(w,x,y,z)$ is given by Table 3

where the variable $\displaystyle b$ is a stochastic parameter defined by

$$b=\begin{cases}1&\text{(with probability $\displaystyle\alpha$)}\\ 0&\text{(with probability $\displaystyle 1-\alpha$).}\end{cases}$$

(8)

The motion rule of this stochastic system is as follows.

• •

  An isolated particle (010) moves to its neighboring right empty site.

• •

  For a pair of adjacent two particles (0110), both stay at their positions.

• •

  For a sequence of more than two particles ($\displaystyle 011\ldots 10$), the rightmost particle moves to its neighboring right empty site with probability $\displaystyle\alpha$. Particles other than the rightmost stay at their positions.

Figure 6 shows an example of time evolution of the transformed stochastic system. From the above motion rule of particles, the mean flux of the system can be expressed by the local densities.

$$Q=\lim_{n\to\infty}{\left(\alpha\rho_{1110}^{n}+\rho_{010}^{n}\right)}.$$

(9)

Let us consider the stochastic process provided by Table 3. Define a ‘configuration’ by a set of values over all sites at any time. Then it changes to another along the time evolution by Table 3. This transition of configurations can be treated as a stochastic process using the transition matrix describing the probability of their transition. This transition process can be separated by sets of configurations since all of them cannot be obtained by the evolution of a given configuration. Thus, we define an irreducible set $\displaystyle\Omega$ of configurations where any of $\displaystyle\Omega$ can be transferred to all other ones. Moreover, to make the expression of $\displaystyle\Omega$ compact, we identify configurations up to cyclic rotation.

Note that $\displaystyle\Omega$ cannot be determined only by length $\displaystyle L$, two conserved quantities $\displaystyle m_{1}$ and $\displaystyle m_{110}$. For the same set of $\displaystyle L$, $\displaystyle m_{1}$ and $\displaystyle m_{110}$, there exist configurations which cannot be transferred each other. For example, for $\displaystyle L=10,m_{1}=6,m_{110}=2$ there are two irreducible sets $\displaystyle\Omega_{1}$ and $\displaystyle\Omega_{2}$ given by

$$\Omega_{1}=\begin{array}[]{l}\{0001101111,\ 0001110111,\ 0001111011,\ 0010110111,\\ \ 0010111011,\ 0011011101,\ 0011101101,\ 0101011011\},\end{array}$$

$$\Omega_{2}=\begin{array}[]{l}\{0011001111,\ 0011010111,\ 0011100111,\ 0011101011,\ 0101101011\}.\end{array}$$

The transition matrix of $\displaystyle\Omega_{1}$ is

$$\begin{array}[]{c}0001101111\\ 0001110111\\ 0001111011\\ 0010110111\\ 0010111011\\ 0011011101\\ 0011101101\\ 0101011011\end{array}\left(\begin{array}[]{cccccccc}1-\alpha&0&0&0&0&\alpha&0&0\\ (1-\alpha)\alpha&(1-\alpha)^{2}&0&0&0&\alpha^{2}&(1-\alpha)\alpha&0\\ 0&\alpha&1-\alpha&0&0&0&0&0\\ 0&1-\alpha&0&0&0&0&\alpha&0\\ 0&\alpha&1-\alpha&0&0&0&0&0\\ 0&0&0&1-\alpha&0&0&0&\alpha\\ 0&0&0&\alpha&1-\alpha&0&0&0\\ 0&0&0&0&1&0&0&0\\ \end{array}\right),$$

where each component of the transition matrix ($\displaystyle a_{i,j}$) is a transition probability from the $\displaystyle i$ th configuration to the $\displaystyle j$ th configuration which is determined by the local dynamics. An eigenvector of the matrix for eigenvalue 1 is

$$(\frac{1-\alpha}{\alpha^{2}},\ \frac{1}{\alpha^{2}},\frac{1-\alpha}{\alpha^{2}},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ 1).$$

Supposing that the time evolution is ergodic, the above eigenvector shows ratio of probability of which each configuration occurs in the asymptotic state. The transition matrix of $\displaystyle\Omega_{2}$ is

$$\begin{array}[]{c}0011001111\\ 0011010111\\ 0011100111\\ 0011101011\\ 0101101011\\ \end{array}\left(\begin{array}[]{ccccc}1-\alpha&0&0&\alpha&0\\ 1-\alpha&0&0&\alpha&0\\ 0&2\alpha(1-\alpha)&(1-\alpha)^{2}&0&\alpha^{2}\\ 0&\alpha&1-\alpha&0&0\\ 0&0&1&0&0\\ \end{array}\right),$$

and its eigenvector of eigenvalue 1 is

$$(\frac{2(1-\alpha)}{\alpha^{2}},\ \frac{2}{\alpha},\frac{1}{\alpha^{2}},\ \frac{2}{\alpha},\ 1).$$

We derive an example of eigenvector for another case of $\displaystyle L=16,m_{1}=11,m_{110}=4$.
One of sets of configurations is

$$\begin{array}[]{l}\{0011011011011111,\ 0011011011101111,\ 0011011011110111,\\ \ 0011011011111011,\ 0011011101101111,\ 0011011101110111,\\ \ 0011011101111011,\ 0011011110110111,\ 0011011110111011,\\ \ 0011011111011011,\ 0011101101101111,\ 0011101101110111,\\ \ 0011101101111011,\ 0011101110110111,\ 0011101110111011,\\ \ 0011101111011011,\ 0011110110110111,\ 0011110110111011,\\ \ 0011110111011011,\ 0011111011011011,\ 0101101101101111,\\ \ 0101101101110111,\ 0101101101111011,\ 0101101110110111,\\ \ 0101101110111011,\ 0101101111011011,\ 0101110110110111,\\ \ 0101110110111011,\ 0101110111011011,\ 0101111011011011\}.\end{array}$$

The eigenvector for eigenvalue 1 of the transition matrix is

	$\displaystyle\displaystyle(\frac{1-\alpha}{\alpha},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ \frac{1-\alpha}{\alpha},\ \frac{1}{\alpha},\ \frac{1}{(1-\alpha)\alpha},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ \frac{1-\alpha}{\alpha},\ \frac{1}{\alpha},\ \frac{1}{(1-\alpha)\alpha},\ \frac{1}{\alpha},\ \frac{1}{(1-\alpha)\alpha},\$
	$\displaystyle\displaystyle\frac{1}{(1-\alpha)\alpha},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ \frac{1}{\alpha},\ \frac{1-\alpha}{\alpha},\ 1,\ \frac{1}{1-\alpha},\ 1,\ \frac{1}{1-\alpha},\ \frac{1}{1-\alpha},\ 1,\ \frac{1}{1-\alpha},\ \frac{1}{1-\alpha},\ \frac{1}{1-\alpha},\ 1).$

Examining other concrete examples, we propose a conjecture for the asymptotic distribution of configurations as follows.

Conjecture: For arbitrary $\displaystyle\Omega$, the probability of configuration $\displaystyle x\in\Omega$ in the asymptotic state is given by

$$p\left(x\right)\propto\frac{\alpha^{m_{010}\left(x\right)}}{\left(1-\alpha\right)^{m_{1110}\left(x\right)+m_{010}\left(x\right)}},$$

(10)

where $\displaystyle m_{1110}\left(x\right)$ and $\displaystyle m_{010}\left(x\right)$ is the number of local patterns 1110 and 010 in the configuration $\displaystyle x$.

By the above conjecture, the probability of $\displaystyle x$ in the asymptotic state is

$$p\left(x\right)=\frac{\frac{\alpha^{m_{010}\left(x\right)}}{\left(1-\alpha\right)^{m_{1110}\left(x\right)+m_{010}\left(x\right)}}}{\sum_{k_{1},k_{2}}{\frac{\alpha^{k_{2}}}{\left(1-\alpha\right)^{k_{1}+k_{2}}}N\left(k_{1},k_{2}\right)}},$$

(11)

where $\displaystyle N(k_{1},k_{2})$ is a partition function, that is, the number of configurations $\displaystyle x$ satisfying $\displaystyle m_{1110}(x)=k_{1}$ and $\displaystyle m_{010}(x)=k_{2}$. We obtain that the maximum value of $\displaystyle k_{1}+k_{2}$, that is, the maximum value of the sum of numbers of local patterns 1110 and 010 in $\displaystyle\Omega$, is $\displaystyle\min{\left(L-m_{1},m_{1}-2m_{110}\right)}$[9]. Since the mean flux $\displaystyle Q$ is an expected value of $\displaystyle\lim_{n\to\infty}{\left(\alpha\rho_{1110}^{n}+\rho_{010}^{n}\right)}$, we have

$$Q=\frac{1}{L}\frac{\sum_{0\leq k_{1}+k_{2}\leq\min{\left(L-m_{1},m_{1}-2m_{110}\right)}}{\left(\alpha k_{1}+k_{2}\right)\frac{\alpha^{k_{2}}}{\left(1-\alpha\right)^{k_{1}+k_{2}}}}N\left(k_{1},k_{2}\right)}{\sum_{0\leq k_{1}+k_{2}\leq\min{\left(L-m_{1},m_{1}-2m_{110}\right)}}{\frac{\alpha^{k_{2}}}{\left(1-\alpha\right)^{k_{1}+k_{2}}}}N\left(k_{1},k_{2}\right)}.$$

(12)

Using inverse transformation $\displaystyle u_{n-j}^{n}=v_{j}^{n}$, the mean flux of (5) is $\displaystyle\rho_{1}-Q$. Figure 7 shows a comparison between theoretical values by (12) and numerical values.

Since the terms $\displaystyle\alpha^{k_{2}}/\left(1-\alpha\right)^{k_{1}+k_{2}}$ of maximum $\displaystyle k_{1}+k_{2}$ remains in the limit $\displaystyle\alpha\to 1$ for (12), we have

	$\displaystyle\displaystyle\lim_{\alpha\to 1}(\rho_{1}-Q)$	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle\frac{m_{1}}{L}-\frac{1}{L}\frac{\min{\left(L-m_{1},m_{1}-2m_{110}\right)}\sum_{k_{1}+k_{2}=\min{\left(L-m_{1},m_{1}-2m_{110}\right)}}{N\left(k_{1},k_{2}\right)}}{\sum_{k_{1}+k_{2}=\min{\left(L-m_{1},m_{1}-2m_{110}\right)}}{N\left(k_{1},k_{2}\right)}}$
		$\displaystyle\displaystyle=$	$\displaystyle\displaystyle\frac{m_{1}-\min{\left(L-m_{1},m_{1}-2m_{110}\right)}}{L}=\rho_{1}-\min{\left(1-\rho_{1},\rho_{1}-2\rho_{110}\right)}$
		$\displaystyle\displaystyle=$	$\displaystyle\displaystyle\rho_{1}+\max{\left(\rho_{1}-1,2\rho_{110}-\rho_{1}\right)}=\max{\left(2\rho_{1}-1,2\rho_{110}\right)}.$

It coincides with (4) of the deterministic system (1).

We analyzed asymptotic behavior of the stochastic particle system of 5 neighbors given by (5) and (7). Considering the irreducible sets of configurations, we proposed a conjecture about components of eigenvector of transition matrix for eigenvalue 1 is determined only by the number of the local patterns 1110 and 010 depending on each configuration. Using this conjecture, we derived asymptotic distribution of configurations. Then, we derived mean flux of the stochastic system as an expected value of the densities of local patterns 1110 and 010. The three-dimensional fundamental diagram can be obtained theoretically by (12), and it coincides with the numerical results. Moreover, we derived theoretical formula of mean flux of the deterministic system through the limit $\displaystyle\alpha\to 1$.

The partition function $\displaystyle N(k_{1},k_{2})$ is the number of configurations of which the number of local patterns 1110 and 010 are $\displaystyle k_{1}$ and $\displaystyle k_{2}$. Since it is hard to enumerate configurations with specific local patterns for arbitrary set $\displaystyle\Omega$, general formula of the partition function has not been obtained yet. However, once $\displaystyle\Omega$ is given concretely, $\displaystyle N(k_{1},k_{2})$ can be easily calculated, thus we confirmed the theoretical formula of the fundamental diagram. Derivation of the general formula of the partition function is one of the future problems. Moreover, another future problem of our theory is to prove the configuration of the asymptotic distribution (10).

Partition function and fundamental diagram are generally hard to derive for arbitrary stochastic particle systems. Our results reported in this article imply a breakthrough to solve this difficulty. The key idea is to introduce two or more quantities for classification of asymptotic set of configurations and to derive the distribution using them. Since this idea can be applied to other systems, it is interesting to obtain a general formulation of stochastic particle systems with a common mechanism.