Analytic Solution of an Active Brownian Particle in a Harmonic Well Michele Caraglio1and Thomas Franosch1 1Institut f ur Theoretische Physik Universit at Innsbruck

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Analytic Solution of an Active Brownian Particle in a Harmonic Well

Michele Caraglio1and Thomas Franosch1

1Institut f¨ur Theoretische Physik, Universit¨at Innsbruck,

Technikerstraße 25/2, A-6020 Innsbruck, Austria

(Dated: October 11, 2022)

We provide an analytical solution for the time-dependent Fokker-Planck equation for a two-

dimensional active Brownian particle trapped in an isotropic harmonic potential. Using the passive

Brownian particle as basis states we show that the Fokker-Planck operator becomes lower diagonal,

implying that the eigenvalues are unaﬀected by the activity. The propagator is then expressed as

a combination of the equilibrium eigenstates with weights obeying exact iterative relations. We

show that for the low-order correlation functions, such as the positional autocorrelation function,

the recursion terminates at ﬁnite order in the P´eclet number allowing us to generate exact compact

expressions and derive the velocity autocorrelation function and the time-dependent diﬀusion coeﬃ-

cient. The nonmonotonic behavior of latter quantities serves as a ﬁngerprint of the non-equilibrium

dynamics.

It is hard to overstate the role of the harmonic oscil-

lator is physics. Being a paradigmatic model for waves

and vibrational phenomena, it serves as a workhorse in

both classical and quantum physics describing diverse

phenomena such as springs, pendulums, molecular vibra-

tions, acoustic oscillations, laser traps, electromagnetic

ﬁelds in a cavity, and resonant electrical circuits, just to

name a few [1,2]. Any smooth potential can be approxi-

mated by a harmonic potential in the vicinity of a stable

equilibrium point [1] and even advanced tools like second

quantization in quantum ﬁeld theory have their roots in

the mathematics of harmonic oscillations [3].

Active matter and directed motion have come under

the spotlight of several research communities, includ-

ing biology [4–7], biomedicine [8–10], robotics [11,12],

and statistical physics [13–25]. However, notwithstand-

ing more than two decades of scientiﬁc eﬀorts on self-

propelled particles, some basic theoretical aspects have

remained elusive since exactly solvable models of even

single active particles are rare. Generally, in external

conﬁning potentials, the steady-state probability distri-

bution is not known analytically, with the notable excep-

tions of active Brownian particles in channels [26] or sed-

imenting in a gravitational ﬁeld [27], and run-and-tumble

particles in one dimension [28–31]. The complete charac-

terization of the time-dependent probability distribution

for a particle starting with certain initial conditions is

even more challenging. In this case, no analytical ex-

pressions are known for conﬁning potentials, and in free

space only solutions in the Fourier domain have been pro-

vided for single active Brownian particle [32–35] and for

run-and-tumble dynamics [36].

The active Brownian particle (ABP) has become the

minimal paradigm for self-propelled particles and it is al-

ready able to describe with a certain accuracy the prop-

erties of motion of a large fraction of existing microswim-

mers [14,19]. Such active particle can be trapped and

monitored by optical [37] or acoustic [38] tweezers which

are well represented by harmonic potentials. While sim-

ulations of ABPs in a harmonic trap can easily be per-

formed by integrating the Langevin equations of motion,

analytical progress is hindered because, despite the lin-

earity of the restoring force, the problem remains nonlin-

ear due to the constraint that the orientation can merely

rotate. Recent signiﬁcant advance has been achieved by

Malakar et al. [39] for the stationary solution of the asso-

ciated Fokker-Planck equation. They express the steady-

state probability in the form of a power-series expansion

in the P´eclet number, a parameter indicating the relative

importance of active motion compared to diﬀusion. How-

ever, the full time-dependent probability distribution of

an ABP in a harmonic trap still remains elusive.

Here we show that, taking the eigenstates of the pas-

sive Brownian particle as an orthonormal basis and upon

proper ordering of these states, the entire Fokker-Planck

operator becomes lower diagonal. This implies that not

only the ground state but the entire eigenvalue spectrum

of the Fokker-Planck operator remains unaltered when

introducing the activity. These surprising ﬁndings al-

lows us to provide an exact expression for the probabil-

ity propagator of an ABP in a two-dimensional harmonic

well, thus going beyond existing theoretical approxima-

tions [40–42] and complementing numerical simulations,

and experiments [38,43]. We also show that exact ex-

pressions of any moment or correlation function can be

readily derived from our solution.

Model. We characterize the overdamped motion of

a two-dimensional ABP in terms of the propagator

P(r, ϑ, t|r0, ϑ0) which is the probability to ﬁnd the par-

ticle at position rand orientation ϑat lag time tgiven

the initial position r0and orientation ϑ0at time t= 0.

Its time evolution is provided by the Fokker-Planck equa-

tion [44]

∂tP=∇·(µkrP) + D∇2P+Drot∂2

ϑP−vu·∇P,(1)

in short ∂tP= Ω Pwith Ω the Fokker-Planck opera-

tor and the formal solution of the propagator is thus

arXiv:2210.04205v1 [cond-mat.soft] 9 Oct 2022

P(r, ϑ, t|r0, ϑ0) = eΩtδ(r−r0)δ(ϑ−ϑ0). The ﬁrst term

on the r.h.s. of Eq. (1) describes the drift motion due

to the harmonic potential U(r) = kr2/2 with spring con-

stant k > 0, whereas µis the mobility of the particle.

The second term encodes the translational diﬀusion with

diﬀusion coeﬃcient D. The ratio D/µ =kBTintroduces

an eﬀective temperature which for a passive particle cor-

responds to the temperature of the solvent. The rota-

tional diﬀusion of the ABP is described by the third term

with rotational diﬀusion coeﬃcient Drot, while the last

term corresponds to the self-propulsion of the particle

with ﬁxed velocity valong the orientation of the parti-

cle, u= (cos ϑ, sin ϑ). In the case of a passive particle,

v= 0, the equilibrium distribution corresponds to the

Boltzmann distribution peq(r, ϑ)∝e−U(r)/kBT, or, with

proper normalization Rdrdϑ peq(r, ϑ) = 1,

peq(r, ϑ) = exp−r2/2d2

4π2d2,(2)

where d:= pkBT/k is the thermal oscillator length. In

particular, the translational and orientational degrees of

freedom are decoupled.

Theory. The Fokker-Planck operator Ω in Eq. (1) ap-

pears to be non-Hermitian already in equilibrium, v= 0.

However, in this case it can be made manifestly Hermi-

tian by a gauge transformation [44]. Here we circumvent

this detour and deﬁne a new operator Lby splitting oﬀ

the equilibrium density

Ω [peq(r, ϑ)ψ(r, ϑ)] =: peq(r, ϑ)Lψ(r, ϑ),(3)

where ψ(r, ϑ) is an arbitrary function depending on the

coordinates rand ϑonly. Then Lcan be naturally de-

composed

L=L0+ Pe L1,(4)

into an equilibrium contribution L0and the non-

equilibrium driving L1, where Pe := vd/D denotes the

P´eclet number and in the following will act as an expan-

sion parameter. In polar coordinates r=r(cos ϕ, sin ϕ)

the equilibrium operator is expressed as

L0ψ=D

r∂r(r∂rψ) + D

r2∂2

ϕψ+Drot∂2

ϑψ−Dr

d2∂rψ ,

(5)

while the active part reads

L1ψ=D

dï−cos(χ)∂rψ−1

rsin(χ)∂ϕψ+r

d2cos(χ)ψò,

(6)

where χ:= ∠(u,r) = ϑ−ϕabbreviates the relative angle

between orientation and position.

Then one readily shows that the equilibrium operator

L0is Hermitian, hφ|L0ψi=hL0φ|ψi, with respect to the

Kubo scalar product

hφ|ψi:= ZdrZ2π

dϑ peq(r, ϑ)φ(r, ϑ)∗ψ(r, ϑ),(7)

and correspondingly its eigenvalues are real and left and

right eigenfunctions coincide. The solution the Hermitian

eigenvalue problem of the equilibrium reference system,

L0ψ=−λψ , (8)

is obtained by a separation ansatz following precisely the

steps of the 2D isotropic harmonic oscillator in the quan-

tum case [45,46] augmented by the uncoupled orienta-

tional diﬀusion. Explicitly, the eigenfunctions read

ψn,`,j (r, ϑ)=n!

(n+|`|)! År

d√2ã|`|

L|`|

nÅr2

2d2ãei`ϕei(j−`)ϑ,

(9)

where L|`|

n(x) are the generalized Laguerre polynomi-

als [47]. Here n∈N0and `, j ∈Z. The quantum numbers

(n, `, j) correspond to the 3 degrees of freedom (r, ϕ, ϑ)

in polar coordinates. The associated eigenvalue is

λn,`,j =1

τ(2n+|`|) + Drot(j−`)2,(10)

with the trap relaxation time τ=d2/D = 1/µk.

Since L0is unchanged under rotations of the position

or the orientation of the particle it commutes with the

corresponding generators L=−i∂ϕand S=−i∂ϑwhich,

borrowing a quantum language, we refer to as ‘orbital

momentum’ and ‘spin’. The eigenfunctions ψn,`,j are

simultaneous eigenfunctions to orbital momentum and

spin with eigenvalues `and s:= j−`. For the active

particle L=L0+ Pe L1remains invariant only under a

simultaneous rotation of position and orientation, such

that the total ‘angular momentum’ J=L+Sis con-

served. Hence, in the full problem jwill be still a good

quantum number.

Note that the eigenfunctions of the equilibrium ref-

erence system are orthonormalized with respect to the

Kubo scalar product (7)

hψn0,`0,j0|ψn,`,j i=δj,j0δ`,`0δn,n0,(11)

and fulﬁll the completeness relation,

peq(r, ϑ)

∞

n=0

∞

`=−∞

∞

j=−∞

ψn,`,j (r, ϑ)ψn,`,j (r0, ϑ0)∗

=δ(r−r0)δ(ϑ−ϑ0).(12)

Moving our attention back to the full problem for an

active particle, the formal expression of the propagator

−5 0 5

x/d

−5

y/d

t= 0.2

simulations

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

−5 0 5

x/d

−5

y/d

t= 1.0

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

−5 0 5

x/d

−5

y/d

t= 100.0

0.0000

0.0025

0.0050

0.0075

0.0100

0.0125

0.0150

0.0175

0.0200

−5 0 5

x/d

−5

y/d

numerics

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

−5 0 5

x/d

−5

y/d

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

−5 0 5

x/d

−5

y/d

0.0000

0.0025

0.0050

0.0075

0.0100

0.0125

0.0150

0.0175

0.0200

FIG. 1. Spatial probability distribution at diﬀerent times tstarting with initial condition r0= 4d,ϕ0= 0, and ϑ0=π/2.

Comparison between simulations and numerics for Pe = 4 and Drot τ= 0.8. For the simulations, statistics has been collected

from 2 ·105independent realizations of the process.

allows us to write

P(r, ϑ, t|r0, ϑ0) = eΩtδ(r−r0)δ(ϑ−ϑ0)

=peq(r, ϑ)X

n,`,j eLtψn,`,j (r, ϑ)ψn,`,j (r0, ϑ0)∗

=peq(r, ϑ)X

n,`,j hrϑ|eLt|ψn,`,j ihψn,`,j |r0ϑ0i,(13)

where, going from the ﬁrst to the second line, we used

Eqs. (12) and (3) and, in the third line, we rely on

Dirac’s bra-ket notation where the isomorphism between

|ψiand ψ(r, ϑ) is made explicit by introducing general-

ized position/orientation states |rϑisuch that ψ(r, ϑ) =

hrϑ|ψi[48]. Then, exploiting twice the identity relation

n,`,j |ψn,`,j ihψn,`,j |=1,(14)

Eq. (13) can be ﬁnally recast in

P(r, ϑ, t|r0, ϑ0)

=peq(r, ϑ)X

n,`,j

Mn,`,j (r0, ϑ0, t)ψn,`,j (r, ϑ),(15)

with

Mn,`,j (r0, ϑ0, t) := hψn,`,j |eLt|r0ϑ0i.(16)

Note that the functions Mn,`,j (r0, ϑ0, t) depend only on

time tand on the initial conditions (r0, ϑ0), which greatly

simpliﬁes the numerical implementation.

To make further progress we rely on the renowned

Dyson equation, familiar from quantum theory [2], for

the time evolution operator

eLt=eL0t+ Pe Zt

ds eL0(t−s)L1eLs,(17)

which can be inserted in Eq. (16), together with the iden-

tity (14), to obtain a useful integral relation for the func-

tions Mappearing in the propagator

Mn,`,j (r0, ϑ0, t) = e−λn,`,j thψn,`,j |r0ϑ0i+

+PeZt

dsïe−λn,`,j (t−s)

×X

n0,`0,j0hψn,`,j |L1|ψn0,`0,j0iMn0,`0,j0(r0,ϑ0,s)ò.(18)

For active particles, Pe >0, the operator L1introduces

couplings between the eigenstates |ψn,`,j i. Starting from

Eqs. (6) and (9), one readily obtains (see also Ref. [39]

for a comparison to the steady-state solution)

L1|ψn,`,j i=

√2τ











√n+`+1 |ψn,`+1,j i−√n+1 |ψn+1,`−1,j iif ` > 0,

√n+1 |ψn,`+1,j i+√n+1 |ψn,`−1,j iif `=0,

√n−`+1 |ψn,`−1,j i−√n+1 |ψn+1,`+1,j iif ` < 0.

(19)

As anticipated, the action of the operator L1does not

modify the quantum number j. Furthermore, its nature

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AnalyticSolutionofanActiveBrownianParticleinaHarmonicWellMicheleCaraglio1andThomasFranosch11InstitutfurTheoretischePhysik,UniversitatInnsbruck,Technikerstrae25/2,A-6020Innsbruck,Austria(Dated:October11,2022)Weprovideananalyticalsolutionforthetime-dependentFokker-Planckequationforatwo-dimensionala...

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Analytic Solution of an Active Brownian Particle in a Harmonic Well Michele Caraglio1and Thomas Franosch1 1Institut f ur Theoretische Physik Universit at Innsbruck

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