Analyzing the impact of a discounted parameter on the reduction of collision time in Brownian particle trajectories

Abstract

Using a set of nanosensors, we propose to study the existence of a finite tracking plan to detect a d-dimensional Brownian particle in the fluid. Each nanosensor moves iteratively along its path, alternating direction at random distances and velocities in both directions from its line’s origin. These distances and velocities are expressed in terms of independent random variables that have probability density functions (PDFs) that are known. We further analyze the density of the random distances in our model by using the Fourier–Laplace transformation. Uncertainty is increased by the potential for a particle to collide with one of the nanosensors and the random sequence of turning points on each line. To account for this uncertainty, we might consider the search distance as a function of a discounted effort-reward parameter. The results of this analysis will give us the prerequisites needed to create a finite expectation value for the particle’s first collision time with one of the nanosensors.

Introduction

Plasma is an important part of physics, and is found in the universe between stars and galaxies, and constitutes a large percentage of cosmic matter. It has unique properties, including that plasma is used to etch computer chips to produce electrical circuits and make semiconductors. A fluid system is microscopic if the collision between particles is high enough that the plasma velocity distribution is close to the Maxwell–Boltzmann distribution law. In the case of collisionless plasma, describing the particle velocity distribution mathematically for each point in the plasma is necessary. So that tracking charged particles in plasmas that transport in space with d-dimensional Brownian motion is one of the numerous applications of search theory. The searchers go at known velocities and distances in most searching techniques. If the search space is a line, the searcher’s goal is to find the target on the left or right portion of the initial point, where it can quickly change course. See El-Hadidy et al.^1–3, El-Rayes et al.⁴, El-Hadidy and Abou Gabal^5,6, Fristedt and Heath⁷ for the majority of methods that have been researched for the line that deal with deterministic distances and velocities.

In order to locate the two-dimensional randomly moving targets on the plane, El-Hadidy et al.^8–10 offered more intriguing search techniques with deterministic distance and regular fixed velocity. El-Hadidy et al.^11–16 used numerous searchers to investigate various methods with deterministic distances and velocities on the space. Obtaining the circumstances that make the first encounter time between a searcher and a moving target finite is the main goal of these previous efforts. However, once the target has been discovered, a number of various search algorithms with deterministic distances and velocities have been discussed in previous studies (Reyniers¹⁷, Balkhi¹⁸, Ohsumi¹⁹, and El-Hadidy et al.^20–25) to find this target in the least amount of time on the line, plane, and space. For more recent and diverse studies on discovering and tracking missing moving and stationary targets on the line, plane and space, readers should review the works El-Hadidy et al.^26–50.

The linear tracking technique handles decision-making under uncertainty by having a set of d nanosensors go back and forth to the origin while the particle location is continuously changing. These procedures can be conceptualized as mathematical models that focus on the best course of action for a decision-maker who must make several decisions with a variety of outcomes throughout time. The expectation value of the first collision time between one of the nanosensors and the d-dimensional Brownian particle must thus be found in this model. Recent studies of passive and active Brownian particles have provided comprehensive analyses of tracking, survival, and first-passage phenomena under various stochastic settings. For instance, Mandal and Ghosh⁵¹ studied the diffusive dynamics of overdamped active ellipsoidal particles in two dimensions and reported anisotropic characteristics in transport behavior. Santra et al.⁵² developed rigorous frameworks for active particle survival probabilities, while relevant research published in the Journal of Chemical Physics accounted for interaction effects and confinement on active particle motion, see Ghosh et al.^53,54. Furthermore, Santra et al.⁵⁵ presented theoretical insights into active Brownian dynamics in two dimensions, offering refined stochastic models. These findings are mostly concerned with motion regularities, confinement, and survival strategies of active particles in continuous space. In contrast, the model proposed in this study incorporates a discounted effort–reward parameter and employs Fourier–Laplace transformations to analytically describe the density of random distances and velocities of nanosensors. This new formulation not only ensures the finiteness of the expected time to collision but also offers an efficient mechanism to reduce search effort under uncertainty, a feature not directly addressed in the previously discussed active Brownian particle models.

We assume that the nanosensors move with an unrestricted manner at random times and locations, covering unpredictable distances and velocities. We propose an analytical expression for the random distance density functions that the nanosensor should perform at random velocities using the Fourier–Laplace transform. We also need to shorten one of these nanosensors’ first collision times with a particle or enhance the chance of one of them detecting the particle. A detailed study would be more reliable if there was some degree of uncertainty in the tracking methodology because many future events pertaining to particle position are unpredictable. Consequently, we could gradually modify the return points of the nanosensors for every line. Therefore, we can use a discount factor to reduce the random distance traveled by the nanosensors as well as the predicted value of the first collision time between one of these nanosensors and the particle. Another interesting way to reduce this predicted value is to use the discount effort reward tracking parameter, which was supplied by Blum et al.⁵⁶ and Alzulaibani and El-Hadidy³⁹. They suggested using this parameter to cut down on the time and expense of figuring out how best to distribute the effort needed to search for a randomly moving target in a dangerous area (a particle in the fluid). Using our model, we can define the discounted effort-reward search as a parameter in the distance function. The minimum expected tracking time found in previous experiments will be shortened by using this technique. The main contribution is showing how the discounted parameter may be used to reduce the predicted timing of the first collision and determine whether a finite tracking strategy is in place.

This paper is organized as follows. Section "Model description" discusses how the model was created. An analytical expression for the probability density function of the random velocities and distances is also provided in this section. In addition, this section shows how the discounted effort-reward track is incorporated into these distances as a parameter. Section "Finiteness of a tracking plan as a function of " discusses the condition that proves our tracking approach exists. In Section "Numerical example", a numerical example was presented, which included a comparison between at different values of the discount effort-reward parameter. Finally, a discussion of the findings wraps up the work.

Model description

We suppose that d number of nanosensors begin their track at the origin on the . The nanosensor here is a random search unit governed by probabilistic rules. These nanosensors discover a set of d lines continually. Every nanosensor moves in both directions from its starting point (origin). We imagine that the nanosensor on the line , respectively has the following strategy: To cut a random distance on the left (right) half of at a given random velocity , start at . Next, return to and travel in order to search ‘s other right (left) half. To discover the left (right) portion of up to , and so on, go back and repeat the previous steps. At track step on each line , the nanosensor moves to a new random point with a random distance , changes the direction and magnitude of its velocity (which can have both positive and negative values to account for the direction of a nanosensor motion), and resumes its search for a new random distance. In our model, we consider the probability density functions (PDFs) of the fundamental and independent random distances and the velocities variables are and respectively. After normalizing to one, each of these PDFs has the formula , and for all , , and is even function (the velocity distribution is symmetric), then the tracking model is bias-free. We model the Brownian particle as a stochastic process where each , is an independent standard Brownian motion in . These independent processes have a drift vector and a covariance matrix that is a diagonal matrix with only non-zero . Also, we know probability distribution for the particle’s initial position.

Random distances with the discounted parameter

As shown in Fig. 1, we allow the nanosensor to shift orientation on at random places in our model. The tracking time of a particle is seen as an effort, as some earlier studies (El-Hadidy et al^27,32,35,36.,) have shown. This time will turn into a random variable if the nanosensor speed and discovery distances are assumed to be random variables.

Fig. 1.

The tracking technique that is used to discover a d-dimensional Brownian particle on.

It is assumed in this study that the probability density functions of random distances and velocities, and , are independent of spatial position, which reflects a homogeneous fluid environment. Consequently, the PDFs remain unchanged under hydrodynamic considerations in the present framework. Nevertheless, the model can be generalized to incorporate spatially varying hydrodynamic effects by allowing the distributions to depend explicitly on position and time , i.e., These PDFs become . Such an extension would only modify the coefficients in the Fourier–Laplace formulation without affecting the overall validity of the analytical framework.

Thus, we can take into account the possibility of effort modeling, as illustrated in Lanillos et al.⁵⁷. Based on the likelihood of detection, they showed how the least amount of tracking effort could be obtained using the largest discounted effort reward (or the reduction parameter); for further details, see Blum et al.⁵⁶. Since the particle can move along one of number of lines in any direction from its starting position, the uncertainty arises after acquiring the return points, which are a series of random locations at any given time t. El-Hadidy et al.’s recent study^39–41 looked at the impact of this discount parameter on the distance function when the nanosensor speed equals one. They have the ability to lower the anticipated value of the particle’s initial collision time with one of the nanosensors. The foundation of this model is the random search effort (time), which is determined by the likelihood of a nanosensors making touch with a particle for the first time, the random traveled distance, and the random nanosensors velocity. Therefore, we can add to the distance function for the covered distance number , on the line , which is the reduction parameter.

d-real lines.

The goal is to ascertain the evolution of the density functions of the locations for the nanosensors , respectively. Let be the PDFs for the initial distribution of the nanosensors at the random points . The variations in the nanosensor’s velocities at the place at time t are represented by PDFs this is known as the frequency of velocity changes.

1

Naturally, the speed varies as the distance between steps changes. Thus, the nanosensor has a velocity . Each nanosensor on the line modifies its velocity at the location , distance , and position for all after completing the random distance at time t. Furthermore, we consider the probability density function of a given velocity as . As a result, we are able to integrate the first term over all of these occurrences on the right side of (1). When the velocities are changed at and the last term of (1) satisfies , it will turn into an impulse function. At any line , and each step if we using the variations in the velocities’ frequencies, then we may express as follows:

2

where is the PDF of non-change velocity till the distance . Hence, changing in velocities will makes to alter does not select a different velocity before to crossing the point . Equations (1) and (2), which can be solved analytically, now describe the motion of on the line with a given initial density and the two PDFs for the random distances and velocities. Therefore, we want to obtain the frequency of velocity changes and then substitute with (2). We get the factor , under the integral from the shift property of the Fourier transformation and when applied it with regard to the spatial coordinate in (1). The Fourier transform of with a reciprocal velocity is obtained by integrating with regard to . Thus, as in Gradshteyn and Ryzhik⁵⁸ and for , the following represents the Fourier transform of (1):

3

where the Fourier components are indicated by the indices and . Let denotes the Laplace component and using the convolution property, and if the Laplace transformation applied with respect to t, then we obtain:

4

where . Thus,

5

where is the frequency velocity variations in the Fourier–Laplace domain. Using the identical procedure, and for ;, the Fourier–Laplace transformation of Eqs. (2) and (5) combined yields:

6

The Fourier–Laplace transform of the exponential functions is more appropriate to obtain our analytical expression, as in Musin⁵⁹. Our analytical expression for , densities and , (nonnegative locally integrable functions on ) with random distances and velocities is therefore sufficient to demonstrate the existence of finiteness for this type of tracking strategy by employing the spatial coordinate in (1).

Influence of the discounted parameter

This model is more flexible, where each nanosensor can move freely on its line with random distances and velocities. We consider the tracking technique which used in our model be where be a combination of continuous functions with random velocity . These functions are considered as the tracking strategies of the d number of nanosensors , where

7

Assume be a random variable that denotes the moment of the nanosensors’s and particle’s first collision time. It is defined by

where is a vector of random variables that reflects the particle’s initial position in and is independent of . We consider the set of all tracking strategies that satisfy (7) and have a velocity (5) for each nanosensor . We look for the condition that provides finiteness for the expected value of (i.e., ), where is the class of all surveying tracking methods. Given that numerous future events related to the particle’s location are unexpected at any given time t, the study approach’s degree of uncertainty would increase the reliability of the findings from a careful investigation. This allows us to manage the return points of each nanosensor along its line in terms of time. We can reduce the expected value of the first collision time between the particle and one of the nanosensors as well as its journey distance by utilizing the significance of the discount parameter.

After the nanosensor cutting the distance (see Fig. 1) on the line , it will become at the random point at time t. In order to ensure that for all we define the sequence: where and , , (constant). This makes the expected value of the random distance and the velocity can be obtained as follows by applying Eq. (5) and (6):

8

where is the expected time to track the distance at the speed . and

9

respectively. Hence, using (8) and (9), the expected value of the time to track is,

10

Hence, the tracking time will be satisfied the following,

11

Then, we have for all where is varies from one nanosensor to another. The primary input consists of determining the condition that results in the tracking plan’s finiteness and minimizing while accounting for the parameter .

Finiteness of a tracking plan as a function of

The first collision can occur at an infinite number of positions in the fluid. This provides us with an infinite set of mutually exclusive, likely collision locations. To investigate the existence of the limited constraints that restrict a tracking plan’s range, we need to define a suitable space. An unconventional approach to integration will be employed for these legal and non-particular categories since the first collision time between one of the nanosensors and the Brownian particle on one of the number of lines at a given value (Legal values) (a vector of random variables that reflects the particle’s position in ) on the supplied field or for each line has a known probability value larger than zero. The particle position, denoted by , is thus defined on a probability space , where is the set of all possible meeting points, is the sigma algebra on these points, and is the probability measure that describes the particle’s location at any given time. An existential tracking strategy can be achieved with the help of the following theorems. They assist us in reducing under the influencing of .

Theorem 1

If the nanosensors move with random velocities (9) and a vector of continuous functions which influenced by the parameter , then is finite if the integrals and are finite.

Proof

Let,

(i) and.

(ii) ,

are assumed to be held. Consequently, employing El-Hadidy’s method⁶⁰, we derive:

this leads to,

Consequently,

this holds . Also, from (ii), we have . Given that the motion of the particle is independent of the motion of the nanosensors (where the moving of the nanosensors is likewise independent), we get a set of mutually exclusive events . Thus, we have . Furthermore, we obtain for every transition step on the line by the nanosensor ,

also,

.

Consequently,

Since are independent events, then we have

.

Let , then

And

Consequently, at any time t, if the searcher speed is finite and , then is finite if the integrals and are finite.

This theorem confirms that will remain bounded under the proposed random tracking strategy. Physically, this means that the search process under random speeds and distances affected by the discount factor will not drift or take an infinite amount of time. The reduced effort-reward factor simulates the fact that nanosensors tend to search less in distant regions over time and focus on closer distances, increasing the probability of early particle detection, which is critical in random environments such as liquids or plasmas.

To show how the discounting factor is sensitive to the conditions that make the foregoing integrals in Theorem 1 truly finite, further scenarios under the impact of any value of should be explored in the following Theorem 2.

Theorem 2

The nanosensors tracking strategy should satisfy: and where and , are linear functions.

Proof

At any step if the particle position holds , for all , then , and if then.

and.

Consequently,

For every , there is a vector of constants . As in El-Rayes et al.⁴, it is possible to obtain, where . Thus,

Let , such that has a normal distribution with mean and variance for each line , where is a sequence of independently random variables. Furthermore, for if , and , then is non-increasing with t (see El-Hadidy¹¹). Thus, if then can be obtained by setting and . We obtain,

when,

and.

where from Feller⁶¹, hold the conditions of the renewal theorem which is bounded for all . Thus, the required is achieved . By the same method, one can prove that .

This Theorem provides structural requirements that ensure the bounded and coherent behavior of the nanosensor trajectories, even amidst uncertainty and randomness. Physically, theory ensures that the nanosensors do not diverge infinitely or become stuck in inefficient loops. It invokes linear bounds to correct the relationship between particle position and nanosensor movement and states a realistic tracking environment that neither degrades search efficiency nor undermines spatial predictability. This enhances the ability of the model to be used in applications such as Nanorobots or plasma diagnostics, where detection must be maintained as constant and efficient as possible despite particle motion variability and noise.

Here, the effort-reward coefficient decreased is an important consideration when modeling and maximizing the localization of a Brownian particle in an intervening fluid medium. The coefficient captures search effort versus expected payoff (i.e., successful particle detection) in a world of uncertainty and stochastics. The importance of this coefficient lies in reducing the expected time to first collision. The coefficient is incorporated into the probability density functions of random distances and velocities used by nanosensors, as shown in Eqs. (1) to (6). This allows for the management of return points and motion dynamics based on effort and uncertainty. This coefficient also helps introduce uncertainty into the mathematical model because, given the stochastic nature of particle motion, it allows prioritizing early detection over long-term effort, making the strategy more efficient and realistic. This parameter also ensures a tracking strategy with a finite expected value. Applying Theorems 1 and 2, we show that appropriate conditions on ensure a finite expected value of the first collision time, and hence the strategy is feasible as well as mathematically valid. This parameter extends the capabilities of typical models by incorporating realistic constraints such as energy consumption, resource utilization, and sensor performance limitations in liquid or plasma mediums.

Numerical example

In this section, we present a numerical study that explores the impact of the discount parameter on the expected first collision time between a nanosensor number and a Brownian particle. We let the random search distances are modelled as exponentially distributed with parameter , where and the search velocities are drawn from a standard normal distribution, To assess the influence of effort-reward discounting, the expected collision time is calculated using the integral: evaluated numerically over a fine grid of and . The results in Table 1 show a consistent decline in as either parameter increases.

Table 1.

Numerical results for discounted collision time.

0.5	0	35.27215
	0.1	24.49455
	0.2	17.99599
	0.3	13.77818
	0.4	10.88647
	0.5	8.818037
	0.6	7.287634
	0.7	6.123637
	0.8	5.217774
	0.9	4.498999
	1	3.919128
1	0	17.63607
	0.1	14.57527
	0.2	12.24727
	0.3	10.43555
	0.4	8.997997
	0.5	7.838255
	0.6	6.889092
	0.7	6.102448
	0.8	5.443233
	0.9	4.885339
	1	4.409019
1.5	0	11.75738
	0.1	10.33364
	0.2	9.153672
	0.3	8.164849
	0.4	7.328009
	0.5	6.613528
	0.6	5.998665
	0.7	5.465726
	0.8	5.000777
	0.9	4.592728
	1	4.232658

To visualize the variation of interaction between the two parameters, we first present a 2D graph as in Fig. 2 that shows the variation of with respect to where every curve is for a distinct fixed value of . Figure 2 highlights how increasing creates an abrupt reduction in the expected collision time, especially for small values. For instance, when , the decay in with is noticeably sharper than in the case of .

Fig. 2.

Expected collision time for different values.

Figure 3 provide a 3D surface plot to illustrate the combined influence of both parameters. The surface confirms the nonlinear relationship: the greatest improvements in expected collision time occur when both and are small. As either parameter grows, the marginal benefit of increasing the other diminishes. This observation is consistent with the intuitive interpretation that a higher (i.e., shorter typical search distances) and a higher (i.e., more aggressive discounting) both serve to restrict search effort near the origin, leading to quicker detection of the randomly moving particle.

Fig. 3.

3D Plot of Expected Collision Time.

Finally, numerical simulations validate the theoretical results that the addition of a discounted effort-reward parameter not only guarantees the finiteness of the mean tracking time but significantly enhances the efficiency of the search process. The coupling between and renders the search policy adjustability easily, which is a viable approach to optimize the deployment of nanosensors in the presence of stochastic uncertainty.

Conclusion and future work

To track a d-dimensional Brownian particle in the fluid, we used a set of nanosensors that move with random distances and velocities on a set of disjoint d-real lines. We employ the Fourier–Laplace transform to provide an analytical expression for the density functions of these distances and velocities. We considered as a vector of independent random variables that indicates the initial particle position. A collection of d non-intersecting real lines is regarded as the search space. We therefore examined the particle tracking distance as a function of the reduced effort reward parameter , accounting for the uncertainty resulting from the chance of the particle colliding with one of the nanosensors and the random sequence of turning points on each line. By performing some analytical equations under the influence of , we were able to demonstrate the existence of a finite tracking strategy and utilize it to prove that (is the expected value of the first collision time). The condition necessary to demonstrate this existentialism is provided by Theorem (1). The requirements that demonstrate that (a combination of vectors of continuous functions where with random velocity ) is finite were also supplied by Theorem (2).

Furthermore, this research indirectly demonstrates a relationship between the proposed stochastic pursuit strategy and the mean square displacement of Brownian particles. By minimizing the expected time to first collision through probabilistic modeling of velocities and distances, the theoretical model is consistent with the fundamental principles used to describe mean square displacement. This means that the reduced effort-reward ratio not only results in improved collision timing but is also associated with the general statistical behavior of particle diffusion in random media.

To find a combination of d-dimensional Brownian moving particles under the influence of the discount parameter, we can develop this model as a generalized model with dependent random distances and velocities with d number of nanosensors in future work.

Author contributions

M. A. E. proposed the idea of the manuscript, put the headline on it, participated in studying the analysis in it, and reviewed it. R. A. wrote the draft of the manuscript and participated in its analysis. M. A. E. and R.A. read and approved the final manuscript.

Funding

This research has no funding.

Data availability

All data generated or analyzed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.