Field Theory of Active Brownian Particles with Dry Friction

We present a field theoretic approach to capture the motion of a particle with dry friction for one- and two-dimensional diffusive particles, and further expand the framework for two-dimensional active Brownian particles. Starting with the Fokker-Planck equation and introducing the Hermite polynomials as the corresponding eigen-functions, we obtain the actions and propagators. Using a perturbation expansion, we calculate the effective diffusion coefficient in the presence of both wet and dry frictions in a perturbative way via the Green-Kubo relation. We further compare the analytical result with the numerical simulation. Our result can be used to estimate the values of dry friction coefficient in experiments.


I. INTRODUCTION
Active matter [1][2][3][4] refers to particles or creatures that can absorb energy from the environment and change their mechanical state by using the energy.The collective behaviors of animals such as swarming [5,6], flocks [7][8][9][10][11], and bacteria [12] can be described effectively by appropriate models.Furthermore, artificial active colloidal particles [13,14] are created experimentally to explore active systems.There are plenty of analytical works from the Stokes-Einstein-Sutherland relation of Brownian motion [15] to active systems [16,17].However, fitting analytical results with experimental ones is always a challenge.For example, the restricted confinement can significantly affect the diffusion coefficient [18].Furthermore, the motion of particles can be influenced by dry friction (Coulomb friction) with the substrate if the size and weight of the particle are large enough.In this case, the buoyancy and thermal fluctuations can barely push the particle away from the substrate.Unlike the wet friction that is always proportional to the velocity, dry friction depends only on the velocity direction.
In the most of the previous studies, dry and wet frictions are not considered together.The particles experiencing dry friction are mostly unaffected by the thermal noise, since the particles are large enough.Moreover, the gravity acting on the particles is usually smaller than thermal noise, and the particles hardly reach the lower surface.Hence, dry friction of such particles can also be neglected.For mesoscopic particles, however, both wet and dry frictions can not be neglected because the gravity drags the particles and their random motions take place at the bottom surface [19,20].A one-dimensional (1D) passive particle with dry friction has been discussed [21][22][23], and further two-dimensional (2D) problem has been also addressed [24].
In this work, we are motivated to explore the motion of an active particle with dry friction.In particular, we obtain the effective diffusion coefficient of a 2D active Brownian particle (ABP) with both dry and wet frictions by using the Doi-Peliti field theory [25][26][27].A 2D ABP moves with a finite velocity, and its director undergoes rotational diffusion [28][29][30].We first introduce the Langevin equations to describe the motion of an ABP, and later convert it to the corresponding Fokker-Planck equation.Since the effective diffusion coefficient can be calculated via the velocity-velocity correlation function by using the Green-Kubo relation [15], we only consider the particle velocity in this work.
In Sec.II, we introduce the mathematical basis of this work, such as the equation of motion and its eigen-system to simplify the calculation.In Sec.III, we use the field-theoretic framework to capture the motion of a 1D diffusing particle with dry friction in a perturbative way.We expand our framework to 2D space in Sec.IV by approximating dry friction to be isotropic.By treating the self-propulsion of an ABP as a perturbative part, we obtain the corresponding effective diffusion coefficient in Sec.V. We also compare the 2D result with the simulation result.A summary of our work and some discussion are given in Sec.VI.

II. MODEL
In this section, we firstly show the mathematical basis of our work.Our main aim is to calculate the effective diffusion coefficient D eff as a function of dry friction coefficient µ.The diffusion coefficient D eff can be extracted from the mean squared displacement in the long-time limit.In this work, we obtain the effective diffusion coefficient by the Green-Kubo relation [15], where the ⟨•⟩ is the ensemble average and d is the space dimension.We describe the motion of an ABP with dry friction by the Langevin equations where ||v|| is the norm of the vector v, w θ (t) = w[cos θ, sin θ] T is the self-propelled velocity of the particle which stays on a ring of a radius of w, and m is the mass of the particle.Furthermore, Γv(t) is the viscous force proportional to the velocity and F v(t)/||v(t)|| is dry friction force whose magnitude is a constant.And 1 2 is a 2D unit matrix.Each component of Ξ and also ζ are Gaussian white noise [31] with zero mean and variance DΓ 2 and D r , respectively, where D and D r are the diffusion and rotational diffusion coefficients respectively.And each component of Ξ has zero correlation with ζ.Finally, x is the position of the particle, and we further define the dry friction term v/||v|| = lim ϵ→0 v/||v + ϵ|| to avoid the zero denominator.
Introducing the rescaled friction coefficients by the mass γ = Γ/m and µ = F/m, we modify the Langevin equation as whose corresponding Fokker-Planck equation is [32,33] Since the diffusion coefficient can be extracted from the velocity-velocity correlation function, we drop the positional variable x in the Fokker-Planck equation, and we only consider the particle in the velocity space.The effective diffusion coefficient of an isolated 2D ABP in free space is well known [34,35], in our case, it presents the zero dry friction limit µ = 0. We write it below for the later comparison The following functions are used later to simplify the calculation, where He n (x) is the n-th order of the probabilist's Hermite polynomials and Ω 2 = Dγ [36,37].Our definition of He n (x) corresponds to 2 where δ n,m is the Kronecker delta, and u n (v) are the eigenfunctions of the operator The velocity-velocity correlation function with the initial velocity v 0 , direction θ 0 at time t 0 can be calculated by where we use v(t) = v and v(t ′ ) = v ′ to simplify the notations and G(v, θ, t|v ′ , θ ′ , t ′ ) is the probability density of finding a particle at velocity v with the self-propulsion direction θ at time t, when the given initial state is at velocity v ′ with the propulsion director θ ′ at time t ′ .Since we choose the stationary state as the initial state, the most right propagator G v ′ , θ ′ , t ′ |v 0 , θ 0 , t 0 = G(v ′ , θ ′ ) does not depend on the initial condition.By using the orthogonality of the Hermite polynomials [39], the above integral can be simplified further, and the result will be presented in the following sections.

III. DIFFUSIVE PARTICLE IN 1D SPACE
We start with the simplest case, i.e., a diffusive particle with dry friction in 1D space.The corresponding Fokker-Planck equation is where in 1D space, the unit vector v/||v|| becomes the sign function v/|v| where |v| is the absolute value of v.We further introduce the notations similar to the 2D case, we define v/|v| = lim ϵ→0 v/|v + ϵ| to avoid the zero denominator.

Action
The corresponding bilinear action and perturbative action are where r is the death rate to maintain the causality, and will be taken to zero after the inverse Fourier transform.We have introduced the annihilation field ϕ and the Doi-shifted creation field φ as [25,27] Here Ω 2 = Dγ, d ¯ω = dω /(2π) and further δ¯(ω) = 2πδ(ω), while u and ũ are consistent with Eq. ( 6).Any expectation value can be calculated perturbatively by a path integral [40] ⟨ where we split the action via Eq.(12a).Expanding the exponential with respect to the perturbative part A pert , we obtain The probability density is the full propagator By plugging the fields Eq. ( 13) into the action Eq.(12b), we obtain where Ωδ n,m comes from the orthogonality relation Eq. ( 7) between u n and u m .The bare propagator can be read off as Similarly, we substitute the fields in Eq. ( 13) into the perturbative action in Eq. (12c), where δ(v) comes from the derivative of the sign function We diagrammatically write the perturbative part of the action as By using the following properties of the Hermite polynomials [36], the analytic form of the above vertex becomes where He n (0) is the "Hermite zero", which is probabilist's Hermite polynomials evaluated at zero with respect to the n-th order.It is trivial to see that which indicates that a propagator does not have an outgoing index 0 unless it is a bare one.Moreover, since the "Hermite zero" of arbitrary odd order is zero, He 2n+1 (0) = 0, we find which shows the outgoing and incoming indices should have the same parity.Otherwise, the corresponding combinations become zero.
In the following, we only consider the perturbative vertices with the limited indices n, m = 0, 1, 2, which are

Propagator
By using the bare propagator and perturbative vertices, the full propagator can be written in a perturbative way as The full propagators is therefore and the corresponding E j are given by We further introduce the recurrence relation between E j and E j+1 The probability density of velocity v at time t with a given initial state v ′ at time t ′ is

Stationary-state correlation function
By taking the limit t 0 → −∞, we obtain the stationary density Previous work shows that taking the stationary limit t 0 → −∞ and the zero death rate limit r ↓ 0 replaces the incoming index by δ m,0 [35], which physically indicates that the steady-state is independent of the initial condition.
Therefore, at stationarity, Eq. ( 30) is , where we have dropped the arguments m, ω, since the observable no longer depends on time and initial state.The form of the stationary density is Since ũ0 (v) does not depend on v, we use the notation ũ0 ≜ ũ0 (v) to indicate the independence of the initial state.We list the first two orders and the recurrence relation of E j as follows Since Λ 0,n is always zero in Eq. ( 24) which vanishes all the propagators with outgoing index n = 0 before taking stationary limit, we define lim n→0 Λ n,m /n = 0 for arbitrary m and perform the summation from q = 1 to avoid the zero denominators in Eq. (35).
Our aim is to use the Green-Kubo relation Eq. ( 1) to calculate the effective diffusion coefficient.Now, we perform the integral of the velocity-velocity correlation function in the present field theory framework where we simplify the notation v = v(t) and v ′ = v(t ′ ).By using Eqs.( 9), ( 31), (34) and the orthogonality of the Hermite polynomials [39], the correlation function becomes We first calculate the time-independent observable in the brackets as where in Eq. (38b) we only consider the finite number of perturbative vertices up to the second order, and O(β 2 ) is used to indicate that we do not consider all the contributions higher than the second order of β.The mathematical details are shown in Appendix A 1. We introduce a dimensionless parameter to simplify the notation.Then we have To calculate the time-dependent observable, we only consider the perturbative part with both incoming and outgoing indices fixed to unity, where Θ is the Heaviside step function, since the integral converges only for t ≥ t ′ .Then, the time-correlation function is obtained by substituting Eqs. ( 40) and (41d) into Eq.(37), By using the Green-Kubo relation in Eq. ( 1) and recalling Ω 2 = Dγ, the effective diffusion coefficient is If there is no dry friction, β → 0, we recover the normal translational diffusion coefficient By expanding the fractions, we obtain the first order correction with respect to the parameter β as

IV. DIFFUSIVE PARTICLE IN 2D SPACE
In 2D space, the corresponding Fokker-Planck equation of a diffusive particle with dry friction is Similar to the 1D case, we split the operator into two parts, Applying the isotropic property of the first operator L 0 , one can simplify the calculation into 1D space with an extra prefactor 2. However, because of the anisotropy of dry friction, there is no such a way to simplify the operator L µ .Hence, we use an approximation of the "sign" function of the velocity where we use the following Hermite expansion of the square root in the denominator, and the prefactor exp (−1/4) is the projection of the dry friction term on the He 0 , with θ = arctan(z).By considering only the zeroth order Hermite polynomial and dropping all the higher order terms, we find the approximation in Eq. ( 47).The operator L µ is rewritten as which is now isotropic.Then, we can simplify this 2D problem to a 1D problem whose corresponding Fokker-Planck equation is where we have introduced an upper index (2) to indicate that the above Fokker-Planck equation is obtained from the 2D equation.The corresponding velocity-velocity correlation function is where the prefactor 2 in the RHS is the dimension factor.The only difference between the Fokker-Planck equation in Eq. ( 50) and 1D case in Eq. ( 10) is an extra prefactor exp (−1/4) in the friction term.Therefore, by following the same steps presented in 1D case in Sec.III, we immediately obtain the velocity-velocity correlation and the effective diffusion coefficient by introducing the modified dimensionless parameter as and they are Here we use an upper index (2) to distinguish the effective diffusion coefficient in 2D from the 1D result in Eq. ( 43), where the parameter β in the 2D problem is changed from the parameter β in the 1D case.We compare our result in Eq. (53b) with the numerical simulation.In this work, only finite types of the perturbation vertices are considered, the friction operator is approximated by Eq. ( 49), and the result is a perturbation calculation using the first three orders of the Hermite polynomials.We compare the effective diffusion coefficient with the numerical simulation and our field theory approach in Fig. 1(a), showing a good agreement between them even β is large.Additionally, the fourth order correction of the time-independent observable ϕ 2 φ0 is also calculated in Appendix A 1.

V. ACTIVE BROWNIAN PARTICLE
ABPs are particles that move with constant speed |w| = w but whose director θ diffuses with the rotational diffusion coefficient D r .The corresponding Fokker-Planck equation of a 2D ABP with dry friction is where w θ = w[cos θ, sin θ] T .Similar to the approach presented in Sec.IV, we first use the approximation of the friction term in Eqs. ( 47) and (49), to approximate the anisotropic operator by an isotropic one.Second, we reduce the problem to 1D and choose the component of v on the x-axis of the Cartesian plane.The Fokker-Planck equation is then with A. Action Accordingly, the bilinear action and perturbative actions are Since there is a new variable θ, the fields in this case are introduced as Here, we use Roman letters for the velocity indices and Greek letters for the director indices.Similar to the 1D case, the corresponding bilinear action and the µ-dependent perturbative action are and the self-motility dependent action A w is There are two different types of perturbative vertices.We diagrammatically write the friction-dependent and self-propulsion dependent vertices in the following way and the bare propagator is

B. Propagator
The full propagator is the summation of all the possible combination of the bare propagator in Eq. ( 62) and the perturbative vertices in Eq. ( 61) = δ¯(ω Similar to Eq. ( 28), we write the full propagator as, where the terms shown in Eq. (63b) are exactly the first three orders of E j (n, m, α, α ′ , ω).We list the recurrence relation as We therefore obtain the probability density of the velocity at v with director θ at time t with the corresponding initial state (v ′ , θ ′ , t ′ ) as

C. Velocity-velocity correlation function
By substituting Eq. (66b) into Eq.( 9), and using the orthogonality of the Hermite polynomials and the exponential terms, we have where the second term comes from the integral over θ ′ .Since we only consider the first three orders of the speed indices n, m = 0, 1, 2 of the perturbation vertices in Eq. (61), only ν = ±1 in the summation are concerned in the second term.We calculate the time-independent observable first by the inverse Fourier transform and take the limits t 0 → −∞ and r ↓ 0 We obtain ϕ 0,0 φ0,0 = Ω , (69a) where the details are presented in Appendix A 2.
For the time-dependent correlation function, we only consider the lowest perturbation vertices, similar to Sec.III.The first one is where β is the corresponding dimensionless parameter introduced in Eq. ( 52).The second term is Then, the velocity-velocity correlation function is By applying the Green-Kubo relation in Eq. ( 1), we obtain the effective diffusion coefficient for an ABP with dry friction as When there is no friction, β = 0, we recover the effective diffusion for an isolated ABP in Eq. ( 5) [34,35,41,42], D Similar to Eq. ( 44), by expanding the fractions, we obtain the first order correction of the effective diffusion coefficient for an ABP with dry friction as We further compare the analytical result in Eq. ( 73) with the numerical simulation in Fig. 1(b).Even with the existence of the self-propulsion, the agreement between the field theoretic approach and the simulation is still good.eff is obtained from the Langevin equation in Eq. ( 3) via the Euler-Maruyama method [43], where the deatils are explained in Appendix B. We have used the following parameters: number of the particles N = 20000, total time T = 400, and time step ∆t = 0.05.The initial state of the simulation is x(−50) = 0 and θ(−50) = 0, where we have spent T0 = 50 to wait for the system to reach the stationary state and then started to average the MSD.

VI. SUMMARY AND DISCUSSION
In this paper, we first discussed the 1D diffusing particle with dry friction.From the velocity-corresponding Fokker-Planck equation in Eq. ( 10), we determined the bilinear and perturbative actions for a diffusive particle with dry friction in Eqs. ( 17) and (19), respectively.The full propagator was derived in Eq. ( 27).Considering the first three orders of the Hermite polynomials, we only count three types of perturbative vertices with non-zero contributions Eq. (26).Therefore, the summation in the propagator is reduced to a single term, and the geometric sum is used to obtain the velocity-velocity correlation function in Eq. (42).Equation (43) was further obtained by using the Green-Kubo relation Eq. (1).
Then, we have extended this framework to the 2D space.For wet friction, we can always reduce the higher dimensional problem to a 1D problem because the system is isotropic.For dry friction, however, such a treatment is no longer possible because it is anisotropic.We then introduced a Hermite expansion of dry friction, and transformed dry friction into an isotropic operator in Eq. ( 47) by considering the leading order.With a prefactor difference, we obtained the 2D effective diffusion coefficient Eq. (53b) from the 1D result in Eq. (43).
Studying the ABP problem is more complicated since the self-propulsion of the particle provides another type of perturbation vertex.By using the same treatment as in the 1D case, we only considered the limited Hermite order (n ≤ 2).Then, we applied the geometric sums to calculate the velocity-velocity correlation function in Eq. ( 72) and further obtained the effective diffusion coefficient in Eq. ( 73).Since we have neglected higher orders of the Hermite expansion for both fields and dry friction operator, the diffusion coefficient is a perturbative result.However, the analytical result in Eq. (73) recovers the numerical simulation very well.
Dry friction force F in the experiment can be estimated by using Eq. ( 73) or the first order expansion in Eq. (75).Substituting back Γ = mγ and F = mµ, and using the Stokes-Einstein-Sutherland relations Γ = 6πηa and D = k B T /Γ [15], where η is the fluid viscosity, a is the radius of the particle, k B is the Boltzmann constant and T is the temperature, we rewrite the dimensionless parameter β as β = e − The present work provides a basis for the characterization of the motion of active matter in velocity space via field theory, especially for mesoscopic particles whose diffusion and mass can not be neglected.Because of the wet friction γ, the calculation is simplified by using Hermite expansion of the fields [37] rather than the Fourier transform [35].This work also provides a basis for ABPs with harmonic interactions.With the aid of the previous work on field theoretic approach of interacting diffusive particles [40,44], we are currently working on ABPs with non-reciprocal harmonic interactions.with a suitable initial condition.In the above, ξ i is a 2D vector and the components are Gaussian random variables with zero expected value and unit variance.Similarly, ζ i are also Gaussian random variables with zero expected value and unit variance.The effective diffusion coefficient is obtained from the mean-squared displacement where M is the total steps and M ∆t = T is the total time, and N is the number of the particles to obtain the ensemble average.

FIG. 1 .
FIG. 1.Comparison between the analytical result and numerical simulation.The horizontal axis is the dimensionless parameter β in Eq. (52), and the vertical axis is the ratio between the effective diffusion coefficient D (2) eff in Eqs.(53b), (73) and the diffusion coefficient D0 in the absence of dry friction in Eq. (74).The numerical result of D (2) particle with a mass m ≈ 10 −15 kg and a radius a ≈ 10 −6 m in water with the room temperature, and assuming F ≈ 0.1 × mg, where g is the gravitational constant, we estimate β ≈ 5 × 10 −4 .If the radius is increased to a ≈ 5 × 10 −6 m, the corresponding parameter becomes β ≈ 0.14, which leads to a substantial decrease of the effective diffusion coefficient.