Non-equilibrium diffusion characteristics of the particles system and its application

In this paper, we discuss a possible generalization of the social influences of the Langevin equation. Using a functional method of the Kramers–Moyal expansion coefficients, we prove a simple rule for the corresponding Fokker–Planck equation for a generalized one-dimensional system driven by associated Gaussian noise. We propose here a generalization of the nonequilibrium behavior of the probability density and probability currents induced by different combinations of parameters. In addition, it is interesting to find that the probability of current transport reversal can be obtained by varying the combination of parameters, but the amplitude of the negative and positive currents is controlled by adjusting the interaction coefficient and the intensity factor of the outfield, respectively.

(a) (b) Fig. 1 Schematic drawing of the cognitive ability, U(x) and −∇U (x i ), of the curl noise imply that some anomalous transports appear in the potential fluctuations for smaller fluctuations and the negative nonlinear mobility at larger bias away from equilibrium [22]. The Lotka-Volterra is used in the presence of two symmetrical non-Gaussian noise sources, which analyzed the time behavior of an ecosystem consisting two competing species and surrounding environment and proved the Levy noise playing an important role in population dynamics by Cognata [23]. It has also been reported in the humans dynamics, which described the self-organized collective effects of opinion formation, group dynamics [24], or other social phenomena [25,26], based on the idea that the Newtonian second law of motion, the ordinary differential equations of the velocity and the acceleration are solved by Helbing and Molnar [27,28]. Koster et al. proposed a simplified model based on Helbing's social force model, preserving the dynamic features expected by many-body system, while effectively solving the stability of this numerical solution [29]. Grauwin [25] based on the continuous opinion evolution model [30] introduced two noises to answer a great challenge raised by sociologists: "Structures that last from nonlasting entities." In terms of research methods, the above work mainly adopts stochastic dynamics simulation to explore the evolution laws of complex multi-particle systems in biological, physics and other disciplines. Therefore, how to use the stochastic dynamics ideas to further study the problem of the transport mechanism caused by noise and human behavior characteristics is a very meaningful work.
In this paper, based on the idea and methods with stochastic dynamics, we investigate the collective dynamical behaviors for the non-monotone external forces, and analyze the universal rules of the statistical characteristics.

The stochastic dynamical model of cognitive ability
How people get useful information from the environment to promote cognitive ability for themselves has always been a hot topic for sociologists, anthropologists and psychologists. Similarly, the ability of Brownian particles to capture useful work from nonequilibrium fluctuations in a periodic structure of spatial symmetry breaking is a hot concern for physicists [31][32][33] in recent years. Generalized stochastic dynamics models are the simplest and efficient way to describe such systems. Of course, the study of this theoretical method for complex human behavior has not been perfect and is worth further study. At present, the methods to study the scaling law of human trajectories have become the main trend in the field in recent years, especially particle physics. Thus, the stochastic dynamical equation [34,35] characterizing the evolution of individual cognitive abilities can be written as: where x i denotes the cognitive ability of individual i, whose value represents the level of cognitive ability, while human cognition is not solitary. From tool use, language and mathematics to beliefs about the world and morality, cognition ability is shaped by learning from social environment, which can be understood as the change of cognitive ability caused by environment force. U (x i ) denotes the corresponding generalized potential of cognitive ability. −∇U (x i ) denotes the negative gradient of potential, which evaluates the tendency as Fig. 1(b) shows, which described the correlation curve between the environmental degradation and the income level of citizen by the environmental Kuznets Curve [36][37][38][39][40]. Thus, the curve of generalized potential is evolved by the gradient force of integration, as Fig. 1(a) shows. They indicate that the behaviors of individual in the high-potential region are active; however, because of the limitation of corresponding low-cognitive, capturing the effective information from the environment will be transformed only a few into improving themselves. With the increased cognitive ability, the conversion efficiency rate is also improved, which induces the driving force to present the nonlinear increased tendency. When the cognitive reaches the determined level, the relation is broken and presents the inversing tendency, which implies that capturing the relative efficiency information decreased so as to induce the driving force to decrease. Therefore, the negative gradient force is selected as −∇U ( , where E denotes the intensity factor of external force. In Eq. 1, r (x i ) denotes the accumulation effect for i − th among others individuals. As is known to all, a particle with similarity or same attribute is easy to represent the resonance phenomenon. Of course, human behavior is no exception. Thus, the individual of similar cognitive ability is easy to tend the consensus in forming opinion because of the higher trust, the greater influence and the higher frequency of communication [41]. At present, there has been a hot topic of the simulated human behaviors by using the hierarchical methods [41][42][43][44][45][46]. With the matured internet technology and the diversity of communication media, the localized property of traditional way no longer exists. Therefore, the correlation effect is only considered the localized individual that they have the similar cognitive level ( x i − x j ≤ π 4 ), but the influence of Euclidean distance is ignored, which is selected as where N denotes the total number of individuals in the group, N i represents the number of individuals that can be associated with the i − th. U (x i , x j ) represents the interaction potential between the individual i and the j, which is selected as the π 4 cycles of sine function that it implies that the relation between the gradient potential and the cognitive ability emerges that the smaller the difference among the individuals, the bigger the interaction obviously effects. r 0 represents the interactive intensity factor. Here, the interaction term is calculated by the local mean field method in the latter.
In Eq. 1, ξ (x i , t) and ζ (t) represent the multiplicative noise induced by the internal random behaviors and the additive noise induced by the fluctuation of environment, respectively. They are the random effects generated by a Gaussian white noise, which correspond to the noises with zero mean, and the autocorrelation function and the inter-correlation, which can be written as: where Dg(x i , y i ) and Q represent the correlation strength coefficients of multiplicative and additive noise, respectively, and y i denotes the cognitive ability of the individual at the t , g(x i , y i ) denotes the arbitrary smooth function, which is selected as g(x i , y i ) (x i + y i )/2. From Eq. 3, we can obtain the autocorrelation function of superposition noises, which can be given by If the number of the investigating object is much enough, the dynamical law be given by [47] x Equation 5 depicts the stochastic kinetic evolution trajectory of individual cognitive ability, its purpose is not to predict the trajectory itself, but interested in the statistical characteristics of the group trajectory. Therefore, we focus our attention from random variables to the probability distribution of groups, which studies the time evolution law of the Fokker-Planck equation corresponding to the stochastic kinetic equation. The drift coefficient and diffusion coefficient of the Fokker-Planck equation are calculated by the Kramers-Moyal expansion coefficient, detailed derivation can be seen in Refs. [34,35,47,48], which can be given by Therefore, the Fokker-Planck equation is written as: Eur. Phys. J. Plus (2022) 137:874 We simultaneously integrate the time on both sides of the Fokker-Planck equation and introduce the probability transition function, making a Fourier integral transform of theδ function obtained by the integral. Therefore, its solution can be written as: To further reveal the non-equilibrium transport phenomenon of noise and spatially associated induced systems, the concept of probability current [49][50][51][52][53][54][55][56] is introduced based on the probability density to investigate its non-equilibrium transport characteristics. According to the mass continuous equation [57], the probability current function obeys the following equation: Correlating with Eq. 8, it can be written as: That is, where r (x, t) denotes derivatives of r(x, t) with respect to x in the first order.

The diffusion characteristic of particles in non-equilibrium system
By investigating a large number of parameters combination, we find, adjusting the intensity factor E and r0 in the determined correlation coefficients of the multiplicative noise and the additive( , that the probability density evolution of the group system under different parameter combinations shows some novel nonequilibrium phenomena. When E < 1.0 × 10 −4 , the probability density of velocity exhibits the equilibrium distribution. Meanwhile, when E > 1.0 × 10 2 , it also tends to the equilibrium distribution. If the intensity factor E is between 1.0 × 10 −4 and 1.0 × 10 2 , it exhibits some novel non-equilibrium phenomena, as shown in Figs. 2, 4 and 6.

The inverse phenomenon of non-equilibrium induced by the noises
In the outfield, the probability density distribution of cognitive ability is shown in Fig. 2. In initial evolution at time step 20, as Fig, 2(a) shows, it presents the equilibrium of Boltzmann distribution, and the platform of distribution at left side gradually moves down and grows larger with time step 40, 60, 80. However, the platform emerges the valley with the accumulated correlation effects between the noise and the external force at time step 83, as Fig. 2(b) shows. As the effects are enlarged, the valley is deepened. Moreover, the right side of near-valley emerges the oscillation phenomena, from the local amplification diagram of Fig. 2(b), the oscillations undergo the processes of enhancement (at step 94) and attenuation (at step 98). And then, the evolution probability shows the inverse phenomenon, as Fig. 2(c) shows, which implies the recovery process that the valley is lifted and the oscillation phenomenon disappears. But the recovery relaxation time is longer than the growing time which can be given by 150 − 104 > 98 − 83, which presents the resilient hysteresis of system. The inverse phenomena indicates that under weak external forces and interaction conditions, the correlation between noise and action forces causes the particles to produce the negative mobility or the negative impedance behaviors [58,59], which forms the recovery process; however, the hysteresis characteristic emerges due to the reduced plasticity before the non-equilibrium behaviors. Via the non-equilibrium, the probability density is still the Boltzmann distribution (the fitting red curve), as Fig. 2(d) shows. For the description of the time evolution of probability, it exhibits that the non-equilibrium behaviors appear due to the correlation effects between the noises and the forces in the puniness outfield, and when the diffusion of the nonequilibrium effect achieved to the determined context, the negative mobility of the particle system induced by the correlation effects forming the inverse phenomena of the non-equilibrium. However, since the nonequilibrium processes of complex open systems are damping behavior, the elastic formation exhibits hysteresis behavior. It is shown that the non-equilibrium phenomenon of public opinion caused by the correlation effects between stochastic and deterministic forces is retarded due to the characteristics of opinion propagation such as hysteresis and irreversibility. Comparing the peak value P Max of Fig. 2(a) with Fig. 2(d), the value of former P Max (x, 10) 0.21 is less than the latter P Max (x, 500) 0.25, which implies that the randomness and the coupling characteristic play the key role of the public opinion in the puniness outfield. If the system experiences the coexistence of multiple opinions, collective behavior loses the directionality of inducing impaired behavior and reducing collective cognitive ability.
The corresponding probability current also presents the similar inverse and non-equilibrium behaviors, as shown in Fig. 3, but it is negative. Here, the negative only denotes the direction. Based on the driving-force of cognitive model, we can regard x 3.14 as Because from the natural law viewpoint the particles are easy to move from the big driven-force to the little. Therefore, the negative probability current exhibits that the pattern of probability density is transported into the center, which is divided into the two parts of from the low-cognitive-level to the high and from the high-cognitive-level to the low. The former means the high-cognitive-level of collection lost the capturing much effective information of dynamics to form the decreased relative cognitive ability. The latter embodies the low-cognitive-level of collection excited by the fluctuation of environment to capture much social resources and promote the effects of self-cognitive. From Fig. 3(a) at the different time step 20, 40, 60, 80, it shows the valley of probability current moves to right side and broads, which indicates the mean and the variance of current enlarge to form the directed probability current with the strong stochastically fluctuation. When the probability current achieved at the extreme value as shown in Fig. 3(b) at time step 82 shown, the right side of valley emerges the local fluctuation at time step 86. And then, the local fluctuation is gradually enlarged, as shown in Fig. 3(b) at time step 90, 94, 98. Because of the negative mobility induced by the correlation effects, the current forms the non-equilibrium of inverse behaviors, as shown in Fig. 3(c) at time step 104, 110, 120, 130, 140, 150. However, the probability current reaches the equilibrium of single valley distribution. Compared with the initializing current, the absolute value of valley is less than the one, which implies the situation of multi-perspective coexistence restrains the evolution of cognitive and enlarges the randomness.

The single bulge of non-equilibrium induced by the correlation between the noises and the forces
To study the effect of the external force on the probability density, we adjust the intensity factor of external force E 4.0 on above setting parameters. In comparison with the pattern of the probability density in Fig. 2, it shows significant differences in Inducing the non-equilibrium behaviors becomes sharp, which can be understood the time evolution of non-equilibrium process becomes shorter than the puniness conditions. 2) The amplitude of non-equilibrium induced by the correlation effects between the noises and the interaction force is enlarged and the modes of probability density become different. There also exist the similarities: Both the initial state and the final of probability density are equilibrium of Boltzmann distribution. Figure 4(a) shows the near-equilibrium distribution of probability density time step 2, 4, 6, 10, whose maximum decrease indicates the average of cognitive level is improved. However, the near-equilibrium distribution is broken at time step 12, which emerges the bulge at position x 2.4 and quickly grows at time step 13. And then, the bulge at time step 14 is transformed into the asymmetries double-peak, which moves to both sides, respectively, as shown in Fig. 4(b). This situation is in accordance with the uncoupled among the individuals [47]. Until step 150, the bulge disappears at the edge and tends to the near-equilibrium, as shown in Fig. 4(c). The process illuminates the new opinion induced by the correlation effects between noises and the cognitive ability, quickly emerging the divergence to form the situation of double-opinion coexistence. Here, the one followed by the low-cognitive individuals, and the other tends to the high-cognitive group. With constant competition and game among opinions, the pattern of double-opinion coexistence disappears and the collective behaviors dominated by a few leaders reforms. Comparing the maximum of probability between Fig. 4(a) and (c), the maximum of final probability is less than the initial, which indicates the mean of group is promoted. As for group's cognitive behaviors, the above non-equilibrium process promotes the improvement of themselves. That is, the non-equilibrium in the condition is beneficial to develop the group and the individuals because of the directionality of the strong outfield factor. The group's panic psychology is induced by the non-equilibrium fluctuation such as the gossip and the public opinion; however, the individuals should draw lessons from the process and establish the ways to analyze the public opinion as the event progresses. Figure 5 shows the patterns of corresponding probability current. Compared with Fig. 3, it is still negative value but the amplitude enlarges, which means that the outfield strengthens the directional current. Figure 5 current, whose minimum enlarges and the distribution flattens with time evolution. It presents the local probability current gradually diffuses and homogenizes, which forms the determined probability current. When the probability current achieves the extreme value, the distribution of probability current is broken, which emerges the valley at position x 2.4 at time step 12 and forms the flow of local single opinion. And then, the valley quickly deepens and transforms into the double valley and moves to both sides at time step 13, 14, 20, as shown in Fig. 5(b), which means the flow of single opinion is transformed into the bidirectional flow. At last, the double valleys disappear until time step 150 to form the near-equilibrium distribution of probability current at position x 0, as shown in Fig. 5(c). Compared with the minimum between Fig. 5(a) at time step 2 and Fig. 5(c) at time step 500, it is bigger indicating the negative flow decrease, which further illustrates the development of group's cognitive described in Fig. 4.

The non-equilibrium behavior induced with the strong interaction
Based on the above parameters' combinations, we adjust the interaction factors to investigate the effect for the evolution model of system. Therefore, r 0 6.0 in Figs. 4 and 6 represents evolution probability density is different. Figure 6(a) shows the similar tendency of probability density with the puniness interaction, but emerging the time of non-equilibrium phenomenon is delayed, which reveals the response of group to environment is postponed and the stability is enhanced by the interaction among individuals. However, the equilibrium is broken at time step 23, as shown in Fig. 6(b), which emerges the bulge at position x 2.4 with the accumulation of correlation effects between the noises and the forces. The bulge grows and a new irregular bulge at position x 0.8 appears, at time step 25, whose amplitude is less than the former. And then, the former's bulge mutates into the double peaks moving to the sides, respectively. Another branch of the peak moves to the left merging the irregular bulge into a smaller one and then move to the left side as Fig. 6(b) at time step 50 shown, and the right branch moves disappears at the right side, as Fig. 6(b) at time step 150 shown, forming the new equilibrium distribution, where the red curve is the Boltzmann distribution fitting. The new perspective and multi-perspective coexist in non-equilibrium phenomena induced by the correlation effects between the noises and the determined forces, which present the multi-locality characteristics of group's patterns, so thus the strong interaction happens. Comparing with the maximum probability between the final state and the initial one indicates that the cognitive distribution of the individual is more concentrated near "0", which reveals that the group's cognitive is affected by the non-equilibrium fluctuation of multi-perspective, and weakens the directionality of the outfield to in individuals' decision-making-hesitancy in judging opinion information. Therefore, the group's cognitive tends to the decay. Compared with Fig. 4, because the strong interaction enhances the stability of collective behaviors, it is not easy to be affected by the fluctuation. But the equilibrium pattern is broken and the multilocality novel phenomenon comes out. With multi-perspective situation disappearing, the group's cognitive reduces. The evolution law illuminates some social phenomena, forming the stabilizing structure of tribes, which is difficult to accept the new opinion and propagate. However, if this structure is broken, it would form coexistence of the multi-opinion and lower the decision-making level. Figure 7 shows the corresponding probability current with strong interaction. Its evolution mode is accorded with the weakened situation at the initial stage, as shown in Fig. 7(a), which presents the equilibrium distribution. When the minimum of probability current achieves at the extreme value, as shown in Fig. 7(b), it emerges the valley of non-equilibrium at position x 2.4 at time step 23, and then, the valley is deepened while the probability at position x 0.8 emerges the irregular valley, which forms the novel phenomenon of double non-equilibrium at time step 25. The former's valley is transformed into the double one and moves to the both sides at time step 28. The left merges the irregular into the smaller one and still moves to the left side, as shown in Fig. 7(b) at time step 35,40,50, and the right branch of the peak disappears at the right edge. It forms the new equilibrium pattern after time step 150, and its minimum value is less than the initial value which reveals non-equilibrium process in promoting the activeness of low-cognitive individuals.
3.4 The pattern curve of probability current with different parameters' combinations Through the above adjustment of different external field strengths and individual coupling strengths, it is found that the correlation between noise and determining forces induces a variety of non-equilibrium singular phenomena and the system produces a negative probability flow. To further understand the nonequilibrium probability flow induced by noise association, we examine the direction (a) (c) (b) Fig. 6 It shows the probability current distribution at different time with the interaction factor r 0 6.0 and size of the probability flow generated in the system by recombining the parameters. Figure 8 shows the detailed process of the position and the size of the probability flow from the positive to the negative with different parameters' combinations. From the description of positive probability current in Fig. 8(a), we find that size of the positive is governed by the interaction factor, whose amplitude value decreases with the increasing interaction factor but its adjusting amplitude is small. Figure 8(b) shows the negative situation with different intensity factors of external force, its amplitude increases as increasing intensity factor, and the corresponding position of valley moves to the left side, which indicates that the probability current in the low-cognitive group is easy to be induced with enhanced external force. From Fig. 8(a) to Fig. 8(b), we can obtain that the direction of probability current is governed by the different parameters' combinations. The results are similarity with the conclusions of Refs. [60,61], which describes that the current characteristics of transport of Brownian particles in two-state-flashing-ratchets, the following size and direction in some physical and biological systems, are governed by the asymmetric parameters and heights of two potentials; however, the main difference is that the negative current amplitude is positively correlated to the intensity factor that determines the parameters.

Conclusion and Discussion
By using the theoretical methods of typical stochastic dynamical model, we detailedly discuss the transport characteristics of non-equilibrium with different intensity of outfield and interaction intensity. Study results indicate that the cognitive behaviors of group follow the universal scaling law induced by the correlation effects to form the novel non-equilibrium phenomena of double-/multi-perspective coexistence. In the puniness outfield and interaction effect, collective behaviors are governed by random forces and interaction force (cohesive forces), thus emerging the evolution of locality fluctuations and the hysteresis inverse phenomena, because cohesive forces cause to maintain patterns of group, the characteristics of plasticity and irreversibility of group's cognitive, and the correlation effects among the noises, which lead to form the negative mobility or negative impedance of collective cognitive behavior. In conditions with the strong outfield and the puniness interaction, the evolution law of cognitive behavior is similar to the uncoupled situation that the interaction factor is given by r 0 0, which forms a double-opinion phenomenon in the cooperative game. When the nonequilibrium disappears, it returns to the pattern of Zipf-Pareto [11] dominated by a few leaders. And the process promotes the group's cognitive. In the strong outfield and interaction effects, because of the correlation effects, the nonequilibrium phenomena of multi-perspective coexist. However, the multilocalized structure, formed by strong coupling effects, causes the phenomena that is delayed and restores the development of group's cognitive. These results also showed the empirical statistical characteristics of multi-opinion coexistence polymorphic systems [62][63][64]. At last, through investigating the different parameters' combinations, the probability current is induced by the non-equilibrium, whose size and direction is dominated by the multi-parameters' combinations. Therefore, the positive current is governed by adjusting the interaction factors, and the negative current is governed by the intensity factor of the outfield.