Detecting a weak electric field in a strong noisy background is always an interesting but hard task. We investigate the collective effects of charged Brownian particles in the presence of a weak electric field together with a strong noisy background field by numerical simulation. We show that the pattern formed by snapshots touched by the Brownian particles on one boundary surface can manifest the direction of an applied weak electric field but will be spoiled if there exists a strong noisy background. Moreover, we find that the influence of noise can be suppressed effectively if the correlations caused by the inter-particle electric interaction are taken into account. This work is expected to provide a new angle of view: correlated random walk plays a constructive role in noise suppression.

## I. INTRODUCTION

The detection of weak fields in a strong noisy background attracts one’s attention in many fields due to its potential applications. For example, weak magnetic field detections have widely been applied in airport security, mineral exploration, orientation in navigation, etc. Note that how to suppress the effect of noisy background is always an interesting but hard task in the detection of weak fields.^{1} So far, although there are some studies on the detection of weak electric or magnetic fields,^{2–4} sensitively detecting the weak fields in a complicated noisy background is still a meaningful topic that is worthwhile to be investigated.

Classical random walk theory is originated from Brown motion,^{5–7} which attracted some attention of many physicists, such as Einstein,^{5,6} Pearson,^{8} and Rayleigh,^{9} in the early days. Recently, several studies have proved that the Stocks–Einstein relation^{10–13} is not applicable when the relative size-range between Brown particles and fluid molecules is not over several hundred times and have further extended the relation in the hard-sphere fluid mode.^{12} Anyway, based on the research development of random walk, some corresponding topics, such as random noise,^{1} random process,^{14,15} and spectral analysis,^{16} have emerged and developed. Nowadays, the model of random walks is widely used in the research areas of economy,^{17} physiology,^{18} ecology,^{19,20} and biology.^{21} We know that the process of random walk has different types. Except for Markovian random walk, a long-range memory random walk with non-Markovian relation^{22–29} has recently aroused a wide discussion among researchers. Some corresponding research showed that the non-Markovian random walk in the most extreme case corresponds to a random process that depends on the whole history of the system. Note that a correlated random walk model has recently been proposed to study the behavior^{30,31} of animals, where the existence of correlation can change the property of random walk.

We know that the random walk can provide an effective model for many research fields; thus, in this manuscript, we apply it to investigate the detection of weak electric field theoretically. We show that the collective particle distribution of random walk can manifest the direction of a steady electric field and further provide a model of correlated random walk to suppress the effect of strong noisy background on the detection of weak electric field. We give the dependence of the particle distribution on the correlation strength and external electric field, which enables one to manifest the direction of a weak electric field through the pattern of the particle distribution in the correlated random walk. Since the correlation between particles can suppress the effect of noise distinctly, our proposal of detecting weak electric fields by a correlated random walk is valid even if the noisy background is strong. Thus, this work is expected to provide a new insight into weak field detection.

## II. EFFECT OF ELECTRIC FIELD ON RANDOM WALK

As shown in Fig. 1, we consider a random walk model of many Brownian particles with the same positive charge in a cubic box, where there is a steady electric field along the *x*-axis direction as well as a stochastic electric field in the region of the cubic box. The top surface of the cubic box is smeared with fluorescent powder and will exhibit a light spot once a particle touches it. Thus, a pattern of light spots can be formed after a long term due to the fact that some particles have a probability to touch the top surface after many walk steps. We know that such a pattern records the particle distribution on the top surface, which can reflect the effect of external electric fields. In order to answer the question of whether one can identify an electric field through a random walk system, we study the random walk process numerically. In the numerical simulation, we take the size of the cubic box *X* = *Y* = *Z* = 100*L*, where *X*, *Y*, and *Z* denote the length of three adjacent edges, respectively, with *L* being the step length of the random walk. At the initial time, all particles are in the center of the cubic box, and then, the initial coordinates are *x*_{0} = 50*L*, *y*_{0} = 50*L*, and *z*_{0} = 50*L*. We assume that there are 20 000 particles in the box, and the simulation process of random walk for each particle will be finished once the particle reaches the top surface of the box or the walk step exceeds *n* so that each particle can randomly walk no more than *n* steps. Additionally, we take the periodical boundary condition along the *x*-axis and *y*-axis directions but the open boundary condition along the *z*-axis direction. Note that in dynamic simulations, periodic boundary conditions are used to reduce the effect of container boundaries on particle motion.^{32,33} In our model, we only consider the distribution of particles in the *z*-axis direction, so we take periodic boundary conditions in the *x*-axis and *y*-axis directions to counteract the influence of other boundaries on the particle distribution.

*E*_{x}throughout the box and its direction is along

*x*-axis. To mimic natural surroundings, we assume that there is always a stochastic electric field

*E*_{t}as a noisy background, which acts on all the particles of the system. The direction of

*E*_{t}changes with time, but its value is almost fixed. The random walk will be further affected by the interaction between Brownian particles and the total electric fields. The effect of the stable and stochastic electric fields on particle motion can be characterized by traction strengths

*K*

_{x}and

*K*

_{t}, respectively, which are proportional to the force between the electric field and the particle,

*D*

_{x}represents the interaction coefficient. The interaction force between the electric fields and the particle is

*F*_{x/t}=

*Q*

*E*_{x/t}. For the case of no particle correlation, the existence of the stable electric field will increase the walk probability of the particle in the direction of

*E*_{x}and decrease it in the opposite direction, i.e.,

*P*

_{Kx}(

*x*) = 1/6 + 1/12

*K*

_{x}and

*P*

_{Kx}(−

*x*) = 1/6 − 1/12

*K*

_{x}. Note that the stochastic electric field can also affect the walk probability and, then, that probability will become

*P*

_{Kxt}(

*i*) =

*P*

_{Kx}(

*x*) + 1/12

*K*

_{t},

*P*

_{Kxt}(−

*i*) =

*P*

_{Kx}(−

*x*) − 1/12

*K*

_{t}. Here,

*i*refers to the direction of

*E*_{t}, which changes randomly in the six directions time by time. In the calculation,

*K*

_{x}and

*K*

_{t}are in the range of 0 and 1. The ratio between the values of the stochastic and steady electric fields is fixed. So, the parameter

*γ*= |

*E*

_{t}/

*E*

_{x}| can be taken as different independent constants.

^{34}

First, we consider an ideal case that the particles are affected by only a steady electric field along the *x*-axis direction without a stochastic electric field. In Fig. 2, we plot the corresponding particle distribution on the top surface for different values of walk step *n*. From Figs. 2(a) and 2(b), we can find that the particle distribution is sparse, which means that a small part of particles can reach the top surface within *n* walk steps if *n* is not large enough. In contrast, from Figs. 2(c) and 2(d), it is easy to find that the number of particles reaching the top surface becomes large with an increase in the walk step *n* and the particles are mainly distributed in the range of 50*L*–100*L* along the *x*-axis direction so that the center of the particle distribution on the top surface deviates toward the positive direction of *x*-axis, of which the reason is that the particles feel the electric field force along the *x*-axis direction. This implies that one can manifest the direction of the steady electric field through the center position of the particle distribution on the top surface in the absence of a stochastic electric field.

Note that the noisy background is inevitable in weak field detection. Thus, in order to investigate whether the noise of electric field can destroy the above phenomenon, we further calculate the particle distribution on the top surface considering a stochastic electric field. In Fig. 3, we plot the particle distribution on the top surface with different values of ratio *γ*. From this figure, we can find that the distribution of particles gradually becomes uniform with an increase in the noise ratio. In Fig. 3(d), we can find that the particle distribution on the top surface of the box is uniform and its center does not deviate to the positive direction of *x*-axis for *γ* = 10. This phenomenon shows that a large stochastic electric field (noisy background) can destroy the deviation of the particle distribution on the top surface [see Fig. 3(d)] because the stochastic electric field can cause chaotic movement of the particle distribution. Thus, it is easy to find that one cannot manifest the direction of the steady electric field when the stochastic electric field is strong.

## III. CORRELATED RANDOM WALK

*Q*refers to the charge of the particles and

*r*refers to the distance between the nearest neighbor particles. Furthermore, the relation between the correlation strength and the particle concentration is given as

*ρ*=

*N*/

*V*refers to the particle concentration with

*N*,

*V*being the number of particles and the volume of the cubic box, respectively. In the correlated random walk process, each particle can encounter only one nearest neighbor particle in a concrete direction with the equal probability in the six directions of the axis. In the presence of particle correlation, the probability of Brownian particles moving in the direction (say,

*j*direction) away from its nearest neighbor particle increases under the effect of coulomb repulsion, i.e.,

*P*

_{K}(

*j*) =

*P*

_{Kxt}+ 2

*K*/3, while it is unmodified in the other directions. Since the correlation strength

*K*= |

**K**| and the traction strength

*K*

_{x}= |

**K**

_{x}| and

*K*

_{t}= |

**K**

_{t}| can be tuned by changing the particle concentration

*ρ*and the strength of electric field

*E*

_{x}and

*E*

_{t}, respectively, we treat

*K*

_{x},

*K*

_{t}, and

*K*as independent and adjustable parameters for convenience in our simulation.

*z*-axis, the hopping probability of the particle along the

*z*-axis direction is 1/6 + 5/6

*K*, and that along the other directions is 1/6 − 1/6

*K*, where the value of the correlation strength

*K*is in the range of 0 and 1. According to our theoretical calculation, the expectation

*E*(

*n*) and variance

*D*(

*n*) values of the

*z*-axis coordinate after

*n*steps are gives as

*z*-axis direction and the particles are assumed to start the random walk process from the origin of the coordinate. Since the probability that each particle encounters its nearest neighbor one is equal, the expectation and variance values of the coordinate of all particles after

*n*steps can be easily written as

*x*

_{0},

*y*

_{0}, and

*z*

_{0}are the initial coordinates of the particles. From Eq. (5), we can see that due to the existence of the correlation along the

*z*-axis direction, the particles have a larger probability to walk along the positive direction of

*z*-axis. The larger the correlation strength, the more obvious such a phenomenon becomes. However, if the correlation along the six directions of the axis exists with the equal probability, the phenomenon disappears, which is confirmed with Eq. (7). Meanwhile, from Eq. (8), we can find that for a given walk step

*n*, the variance of the particle distribution will gradually decrease with an increase in the correlation strength. That means the correlation between the nearest neighbor particles can make the particles gradually concentrate together, which will be confirmed by our following numerical results.

## IV. NOISE SUPPRESSION OF CORRELATED RANDOM WALK

Now, we are in the position to study the noise suppression of the correlated random walk numerically. In the numerical simulation, except for the additional parameter *K* characterizing the correlation strength, the other parameter assumption is just like that for the case without particle correlations. In Fig. 4, we plot the particle distribution on the top surface of the cubic box for the case of *K* = 0.3. From this figure, we can find that the particle distribution on the top surface gathers into a pattern whose center deviates toward the positive direction of the *x*-axis when the strong particle correlations are considered. In particular, comparing the four panels in Fig. 4, we can see that the particle distributions are similar for different values of *γ*, which implies that the effect of the stochastic electric field on the particle distribution is negligible. So, the correlated random walk system we considered is strongly robust against the stochastic electric field when the correlation strength is large enough. That is because the correlation between the nearest neighbor particles suppresses the noise of the electric field. Additionally, in Fig. 5, we plot the relation between the particle distribution on the top surface and the *x*-axis coordinate. From this figure, we can find that under the stable electric field and large correlation strength, the distribution curve of the particles deviates toward the positive direction of the *x*-axis, and a slight aggregation effect occurs at *x* = 74*L* forming a bump. The symmetry axis is about *x* = 56*L* with *L* being the step length of the random walk. Note that such distribution curves for different noise ratios are almost coincident, and it is still valid even if the stochastic field is larger than the steady electric field. The results in Fig. 5 confirm that in Fig. 4 that the particle correlations of the system can suppress the noise of the electric field. Thus, the correlated random walk model is more effective to manifest the steady electric field in a noisy background.

Suppressing the effect of a strong noisy background is always a hard task in the detection of a weak field. Since the above results show that large particle correlations can suppress the effect of a stochastic electric field obviously, it is necessary to investigate the effect of the correlation strength on the particle distribution carefully. The corresponding results are plotted in Fig. 6. From this figure, we can see that for a given noise ratio, the distribution of particles along the *x*-axis will gradually form a distribution peak with an increase in the correlation between the nearest neighbor particles. This implies that large particle correlations can suppress the effect of the stochastic electric field and then increase the dispersion velocity of particles, which agrees with the theoretical results in the absence of electric fields. Additionally, comparing the positions of the peaks in the same panel, it is easy to find that the symmetry-axis of distribution peaks will gradually move to the center of the *x*-axis with an increase in the correlation strength. This is because the effect of the particle correlation dominates the correlated random walk process when the correlation strength is larger than the electric fields. Note that although the peak of the particle distribution moves with the change of the correlation strength, its symmetry-axis is always on the positive direction of the *x*-axis, which manifests the direction of the steady electric field. Further comparing Fig. 6(a) with Fig. 6(c) [or Fig. 6(d) with Fig. 6(f)], where the steady electric fields are different but the correlation strengths and noise ratios are the same, we can find that the particle distribution curve in the range of 50*L* and 100*L* is gradually warped with an increase in the steady electric field, which means that the particle distribution in such a range increases. The reason is that the particle distribution in the range of 50*L* and 100*L* will become larger with an increase in the steady electric field.

Furthermore, we calculate the relationship between the half-width of particle distribution peaks and the correlation strength between particles, as shown in Tables I and II for different noise ratios. Note that, for the case of *K* = 0 and *K* = 0.05, the particle distribution on the top surface does not form peak pattern, so the half-width of distribution peaks cannot be given. From the two tables, we can see that with an increase in the correlation strength between nearest neighbor particles, the half-width of particle distribution peaks gradually becomes narrow even for the case of a high noise ratio. This is because the variance of particle distribution decreases with an increase in the correlation strength, which is consistent with our previous theoretical and numerical analysis. Based on the discussion above, it is easy to find that the large correlation between particles can suppress the effect of a stochastic field and induce the particle distribution on the top surface to form a peak pattern. The position of the distribution peak reflects the direction of the steady electric field, so the correlated random walk can provide a new angle of view to detect weak electric fields in a strong noisy background.

. | K = 0
. | 0.05 . | 0.1 . | 0.3 . | 0.5 . | 0.8 . | 1.0 . |
---|---|---|---|---|---|---|---|

K_{x} = 0.01 | ⋯ | ⋯ | 12.0 | 10.0 | 8.5 | 6.0 | 4.0 |

K_{x} = 0.03 | ⋯ | ⋯ | 11.5 | 10.0 | 9.0 | 5.5 | 4.5 |

K_{x} = 0.05 | ⋯ | ⋯ | 11.5 | 10.5 | 9.5 | 6.0 | 4.0 |

. | K = 0
. | 0.05 . | 0.1 . | 0.3 . | 0.5 . | 0.8 . | 1.0 . |
---|---|---|---|---|---|---|---|

K_{x} = 0.01 | ⋯ | ⋯ | 12.0 | 10.0 | 8.5 | 6.0 | 4.0 |

K_{x} = 0.03 | ⋯ | ⋯ | 11.5 | 10.0 | 9.0 | 5.5 | 4.5 |

K_{x} = 0.05 | ⋯ | ⋯ | 11.5 | 10.5 | 9.5 | 6.0 | 4.0 |

. | K = 0
. | 0.05 . | 0.1 . | 0.3 . | 0.5 . | 0.8 . | 1.0 . |
---|---|---|---|---|---|---|---|

K_{x} = 0.01 | ⋯ | ⋯ | 11.5 | 10.0 | 8.5 | 6.5 | 4.0 |

K_{x} = 0.03 | ⋯ | ⋯ | 12.0 | 10.5 | 8.0 | 6.5 | 4.5 |

K_{x} = 0.05 | ⋯ | ⋯ | 11.5 | 10.5 | 9.0 | 6.0 | 4.5 |

. | K = 0
. | 0.05 . | 0.1 . | 0.3 . | 0.5 . | 0.8 . | 1.0 . |
---|---|---|---|---|---|---|---|

K_{x} = 0.01 | ⋯ | ⋯ | 11.5 | 10.0 | 8.5 | 6.5 | 4.0 |

K_{x} = 0.03 | ⋯ | ⋯ | 12.0 | 10.5 | 8.0 | 6.5 | 4.5 |

K_{x} = 0.05 | ⋯ | ⋯ | 11.5 | 10.5 | 9.0 | 6.0 | 4.5 |

## V. SUMMARY

In summary, we consider a random walk system of charged Brownian particles to detect weak electric fields. We show that the collective distribution of particles can reflect the direction of applied steady electric fields in the absence of a noisy background, but such a collective effect will be destroyed by the strong stochastic field. We further propose a correlated random walk model to suppress the effect of a noisy background in the detection of weak electric fields. Through the numerical simulation, we find that for the case of strong particle correlations, the particle distribution on the top surface of the cubic box can still gather into a pattern and the center of the pattern deviates to the direction of the steady electric field even if the stochastic field is strong. Additionally, we investigate the effect of the particle correlation on the particle distribution and show that for given strong stochastic fields, the particle distribution curve gradually forms a peak and its half-width becomes narrow with an increase in the particle correlations. Note that the position of the particle distribution peak can effectively reflect the direction of the applied steady electric field, which implies that the correlated random walk model is robust against a strong noisy background in the detection of electric fields. That is, one can manifest the direction of a weak electric field in a strong noise background according to the collective particle distribution of the correlated random walk. This work is expected to provide a new insight into the detection of weak electric fields.

## ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0304304) and the NSFC (Grant No. 11935012).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Fei Wan**: Investigation (equal); Methodology (equal); Writing – original draft (lead). **Li-Hua Lu**: Formal analysis (equal); Project administration (equal); Writing – review & editing (equal). **You-Quan Li**: Formal analysis (equal); Project administration (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*N*random walkers

*An Introduction to Econophysics*