We study the effect of resetting on diffusion in a logarithmic potential. In this model, a particle diffusing in a potential U(x) = U0 log |x| is reset, i.e., taken back to its initial position, with a constant rate r. We show that this analytically tractable model system exhibits a series of transitions as a function of a single parameter, βU0, the ratio of the strength of the potential to the thermal energy. For βU0 < −1, the potential is strongly repulsive, preventing the particle from reaching the origin. Resetting then generates a non-equilibrium steady state, which is exactly characterized and thoroughly analyzed. In contrast, for βU0 > −1, the potential is either weakly repulsive or attractive, and the diffusing particle eventually reaches the origin. In this case, we provide a closed-form expression for the subsequent first-passage time distribution and show that a resetting transition occurs at βU0 = 5. Namely, we find that resetting can expedite arrival to the origin when −1 < βU0 < 5, but not when βU0 > 5. The results presented herein generalize the results for simple diffusion with resetting—a widely applicable model that is obtained from ours by setting U0 = 0. Extending to general potential strengths, our work opens the door to theoretical and experimental investigation of a plethora of problems that bring together resetting and diffusion in logarithmic potential.

1.
M.
Luby
,
A.
Sinclair
, and
D.
Zuckerman
, “
Optimal speedup of Las Vegas algorithms
,”
Inf. Process. Lett.
47
,
173
(
1993
).
2.
C. P.
Gomes
,
B.
Selman
, and
H.
Kautz
, “
Boosting combinatorial search through randomization
,” in
AAAI/IAAI 98
(
AAAI
,
1998
), Vol. 98, p.
431
.
3.
A.
Montanari
and
R.
Zecchina
, “
Optimizing searches via rare events
,”
Phys. Rev. Lett.
88
,
178701
(
2002
).
4.
D. S.
Steiger
,
T. F.
Rønnow
, and
M.
Troyer
, “
Heavy tails in the distribution of time to solution for classical and quantum annealing
,”
Phys. Rev. Lett.
115
,
230501
(
2015
).
5.
Ł.
Kuśmierz
,
S. N.
Majumdar
,
S.
Sabhapandit
, and
G.
Schehr
, “
First order transition for the optimal search time of Lévy flights with resetting
,”
Phys. Rev. Lett.
113
,
220602
(
2014
).
6.
Ł.
Kuśmierz
and
E.
Gudowska-Nowak
, “
Optimal first-arrival times in Lévy flights with resetting
,”
Phys. Rev. E
92
,
052127
(
2015
).
7.
S.
Reuveni
, “
Optimal stochastic restart renders fluctuations in first passage times universal
,”
Phys. Rev. Lett.
116
,
170601
(
2016
).
8.
A.
Pal
,
A.
Kundu
, and
M. R.
Evans
, “
Diffusion under time-dependent resetting
,”
J. Phys. A: Math. Theor.
49
,
225001
(
2016
).
9.
U.
Bhat
,
C.
De Bacco
, and
S.
Redner
, “
Stochastic search with Poisson and deterministic resetting
,”
J. Stat. Mech.: Theory Exp.
2016
,
083401
.
10.
A.
Pal
and
S.
Reuveni
, “
First passage under restart
,”
Phys. Rev. Lett.
118
,
030603
(
2017
).
11.
A.
Chechkin
and
I. M.
Sokolov
, “
Random search with resetting: A unified renewal approach
,”
Phys. Rev. Lett.
121
,
050601
(
2018
).
12.
M. R.
Evans
and
S. N.
Majumdar
, “
Effects of refractory period on stochastic resetting
,”
J. Phys. A: Math. Theor.
52
,
01LT01
(
2018
).
13.
I.
Eliazar
, “
Branching search
,”
Europhys. Lett.
120
,
60008
(
2018
).
14.
A.
Pal
,
I.
Eliazar
, and
S.
Reuveni
, “
First passage under restart with branching
,”
Phys. Rev. Lett.
122
,
020602
(
2019
).
15.
G. M.
Viswanathan
,
M. G. E.
da Luz
,
E. P.
Raposo
, and
H. E.
Stanley
,
The Physics of Foraging: An Introduction to Random Searches and Biological Encounters
(
Cambridge University Press
,
New York
,
2011
).
16.
A.
Pal
,
L.
Kuśmierz
, and
S.
Reuveni
, “
Home-range search provides advantage under high uncertainty
,” arXiv:1906.06987 (
2019
).
17.
P.
Visco
,
R. J.
Allen
,
S. N.
Majumdar
, and
M. R.
Evans
, “
Switching and growth for microbial populations in catastrophic responsive environments
,”
Biophys. J.
98
,
1099
(
2010
).
18.
D.
Sornette
, “
Critical market crashes
,”
Phys. Rep.
378
,
1
(
2003
).
19.
S.
Reuveni
,
M.
Urbakh
, and
J.
Klafter
, “
Role of substrate unbinding in Michaelis–Menten enzymatic reactions
,”
Proc. Natl. Acad. Sci. U. S. A.
111
,
4391
(
2014
).
20.
T.
Rotbart
,
S.
Reuveni
, and
M.
Urbakh
, “
Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem
,”
Phys. Rev. E
92
,
060101
(
2015
).
21.
A. M.
Berezhkovskii
,
A.
Szabo
,
T.
Rotbart
,
M.
Urbakh
, and
A. B.
Kolomeisky
, “
Dependence of the enzymatic velocity on the substrate dissociation rate
,”
J. Phys. Chem. B
121
,
3437
(
2016
).
22.
T.
Robin
,
S.
Reuveni
, and
M.
Urbakh
, “
Single-molecule theory of enzymatic inhibition
,”
Nat. Commun.
9
,
779
(
2018
).
23.
F.
Wong
,
A.
Dutta
,
D.
Chowdhury
, and
J.
Gunawardena
, “
Structural conditions on complex networks for the Michaelis–Menten input–output response
,”
Proc. Natl. Acad. Sci. U. S. A.
115
,
9738
(
2018
).
24.
É.
Roldán
,
A.
Lisica
,
D.
Sánchez-Taltavull
, and
S. W.
Grill
, “
Stochastic resetting in backtrack recovery by RNA polymerases
,”
Phys. Rev. E
93
,
062411
(
2016
).
25.
I.
Eliazar
,
T.
Koren
, and
J.
Klafter
, “
Searching circular DNA strands
,”
J. Phys.: Condens. Matter
19
,
065140
(
2007
).
26.
I.
Eliazar
,
T.
Koren
, and
J.
Klafter
, “
Parallel search of long circular strands: Modeling, analysis, and optimization
,”
J. Phys. Chem. B
112
,
5905
(
2008
).
27.
S.
Budnar
,
K. B.
Husain
,
G. A.
Gomez
,
M.
Naghibosadat
,
A.
Varma
,
S.
Verma
,
N. A.
Hamilton
,
R. G.
Morris
, and
A. S.
Yap
, “
Anillin promotes cell contractility by cyclic resetting of RhoA residence kinetics
,”
Dev. Cell
49
,
894
(
2019
).
28.
Y. E.
Kim
,
M. S.
Hipp
,
A.
Bracher
,
M.
Hayer-Hartl
, and
F.
Ulrich Hartl
, “
Molecular chaperone functions in protein folding and proteostasis
,”
Annu. Rev. Biochem.
82
,
323
(
2013
).
29.
M. R.
Evans
,
S. N.
Majumdar
, and
G.
Schehr
, “
Stochastic resetting and applications
,”
J. Phys. A: Math. Theor.
53
,
193001
(
2020
).
30.
M. R.
Evans
and
S. N.
Majumdar
, “
Diffusion with stochastic resetting
,”
Phys. Rev. Lett.
106
,
160601
(
2011
).
31.
M. R.
Evans
and
S. N.
Majumdar
, “
Diffusion with optimal resetting
,”
J. Phys. A: Math. Theor.
44
,
435001
(
2011
).
32.
M. R.
Evans
,
S. N.
Majumdar
, and
K.
Mallick
, “
Optimal diffusive search: Nonequilibrium resetting versus equilibrium dynamics
,”
J. Phys. A: Math. Theor.
46
,
185001
(
2013
).
33.
M. R.
Evans
and
S. N.
Majumdar
, “
Diffusion with resetting in arbitrary spatial dimension
,”
J. Phys. A: Math. Theor.
47
,
285001
(
2014
).
34.
C.
Christou
and
A.
Schadschneider
, “
Diffusion with resetting in bounded domains
,”
J. Phys. A: Math. Theor.
48
,
285003
(
2015
).
35.
D.
Boyer
,
M. R.
Evans
, and
S. N.
Majumdar
, “
Long time scaling behaviour for diffusion with resetting and memory
,”
J. Stat. Mech.
2017
,
023208
(
2017
).
36.
A.
Nagar
and
S.
Gupta
, “
Diffusion with stochastic resetting at power-law times
,”
Phys. Rev. E
93
,
060102
(
2016
).
37.
S.
Eule
and
J. J.
Metzger
, “
Non-equilibrium steady states of stochastic processes with intermittent resetting
,”
New J. Phys.
18
,
033006
(
2016
).
38.
A.
Pal
and
V. V.
Prasad
, “
First passage under stochastic resetting in an interval
,”
Phys. Rev. E
99
,
032123
(
2019
).
39.
A.
Pal
,
R.
Chatterjee
,
S.
Reuveni
, and
A.
Kundu
, “
Local time of diffusion with stochastic resetting
,”
J. Phys. A: Math. Theor.
52
,
264002
(
2019
).
40.
A.
Pal
,
Ł.
Kusmierz
, and
S.
Reuveni
, “
Time-dependent density of diffusion with stochastic resetting is invariant to return speed
,”
Phys. Rev. E
100
,
040101
(
2019
).
41.
A.
Pal
,
Ł.
Kuśmierz
, and
S.
Reuveni
, “
Invariants of motion with stochastic resetting and space-time coupled returns
,”
New J. Phys.
21
,
113024
(
2019
).
42.
O.
Tal-Friedman
,
A.
Pal
,
A.
Sekhon
,
S.
Reuveni
, and
Y.
Roichman
, “
Experimental realization of diffusion with stochastic resetting
,” arXiv:2003.03096 (
2020
).
43.
S.
Redner
,
A Guide to First-Passage Processes
(
Cambridge University Press
,
2001
).
44.
A. J.
Bray
,
S. N.
Majumdar
, and
G.
Schehr
, “
Persistence and first-passage properties in nonequilibrium systems
,”
Adv. Phys.
62
,
225
(
2013
).
45.
A.
Pal
, “
Diffusion in a potential landscape with stochastic resetting
,”
Phys. Rev. E
91
,
012113
(
2015
).
46.
A.
Pal
and
V. V.
Prasad
, “
Landau theory of restart transitions
,”
Phys. Rev. Res.
1
,
032001
(
2019
).
47.
S.
Ray
,
D.
Mondal
, and
S.
Reuveni
, “
Péclet number governs transition to acceleratory restart in drift-diffusion
,”
J. Phys. A: Math. Theor.
52
,
255002
(
2019
).
48.
S.
Ahmad
,
I.
Nayak
,
A.
Bansal
,
A.
Nandi
, and
D.
Das
, “
First passage of a particle in a potential under stochastic resetting: A vanishing transition of optimal resetting rate
,”
Phys. Rev. E
99
,
022130
(
2019
).
49.
É.
Roldán
and
S.
Gupta
, “
Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement
,”
Phys. Rev. E
96
,
022130
(
2017
).
50.
D.
Gupta
,
C. A.
Plata
, and
A.
Pal
, “
Work fluctuations and Jarzynski equality in stochastic resetting
,”
Phys. Rev. Lett.
124
,
110608
(
2020
).
51.
A. J.
Bray
, “
Random walks in logarithmic and power-law potentials, nonuniversal persistence, and vortex dynamics in the two-dimensional XY model
,”
Phys. Rev. E
62
,
103
(
2000
).
52.
E.
Martin
,
U.
Behn
, and
G.
Germano
, “
First-passage and first-exit times of a Bessel-like stochastic process
,”
Phys. Rev. E
83
,
051115
(
2011
).
53.
D.
Poland
and
H. A.
Scheraga
, “
Phase transitions in one dimension and the helix—coil transition in polyamino acids
,”
J. Chem. Phys.
45
,
1456
(
1966
).
54.
D.
Poland
and
H. A.
Scheraga
, “
Occurrence of a phase transition in nucleic acid models
,”
J. Chem. Phys.
45
,
1464
(
1966
).
55.
A.
Bar
,
Y.
Kafri
, and
D.
Mukamel
, “
Loop dynamics in DNA denaturation
,”
Phys. Rev. Lett.
98
,
038103
(
2007
).
56.
C.
Fogedby
and
R.
Metzler
, “
DNA bubble dynamics as a quantum coulomb problem
,”
Phys. Rev. Lett.
98
,
070601
(
2007
).
57.
A.
Bar
,
Y.
Kafri
, and
D.
Mukamel
, “
Dynamics of DNA melting
,”
J. Phys.: Condens. Matter
21
,
034110
(
2009
).
58.
V.
Kaiser
and
T.
Novotný
, “
Loop exponent in DNA bubble dynamics
,”
J. Phys. A: Math. Theor.
47
,
315003
(
2014
).
59.
D. A.
Kessler
and
E.
Barkai
, “
Infinite covariant density for diffusion in logarithmic potentials and optical lattices
,”
Phys. Rev. Lett.
105
,
120602
(
2010
).
60.
A.
Dechant
,
E.
Lutz
,
D. A.
Kessler
, and
E.
Barkai
, “
Fluctuations of time averages for Langevin dynamics in a binding force field
,”
Phys. Rev. Lett.
107
,
240603
(
2011
).
61.
A.
Dechant
,
E.
Lutz
,
E.
Barkai
, and
D. A.
Kessler
, “
Solution of the Fokker-Planck equation with a logarithmic potential
,”
J. Stat. Phys.
145
,
1524
(
2011
).
62.
A.
Dechant
,
E.
Lutz
,
D. A.
Kessler
, and
E.
Barkai
, “
Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials
,”
Phys. Rev. E
85
,
051124
(
2012
).
63.
D. A.
Kessler
and
E.
Barkai
, “
Theory of fractional Lévy kinetics for cold atoms diffusing in optical lattices
,”
Phys. Rev. Lett.
108
,
230602
(
2012
).
64.
E.
Lutz
and
F.
Renzoni
, “
Beyond Boltzmann–Gibbs statistical mechanics in optical lattices
,”
Nat. Phys.
9
,
615
(
2013
).
65.
N.
Leibovich
and
E.
Barkai
, “
Aging Wiener-Khinchin theorem
,”
Phys. Rev. Lett.
115
,
080602
(
2015
).
66.
N.
Leibovich
,
A.
Dechant
,
E.
Lutz
, and
E.
Barkai
, “
Aging Wiener-Khinchin theorem and critical exponents of 1/fβ noise
,”
Phys. Rev. E
94
,
052130
(
2016
).
67.
R.
Zwanzig
, “
Diffusion past an entropy barrier
,”
J. Phys. Chem.
96
,
3926
(
1992
).
68.
D.
Reguera
and
J. M.
Rubí
, “
Kinetic equations for diffusion in the presence of entropic barriers
,”
Phys. Rev. E
64
,
061106
(
2001
).
69.
M.
Muthukumar
, “
Polymer escape through a nanopore
,”
J. Chem. Phys.
118
,
5174
(
2003
).
70.
D.
Mondal
,
M.
Das
, and
D. S.
Ray
, “
Entropic resonant activation
,”
J. Chem. Phys.
132
,
224102
(
2010
).
71.
D.
Mondal
,
M.
Das
, and
D. S.
Ray
, “
Entropic noise-induced nonequilibrium transition
,”
J. Chem. Phys.
133
,
204102
(
2010
).
72.
D.
Mondal
, “
Enhancement of entropic transport by intermediates
,”
Phys. Rev. E
84
,
011149
(
2011
).
73.
D.
Mondal
and
D. S.
Ray
, “
Asymmetric stochastic localization in geometry controlled kinetics
,”
J. Chem. Phys.
135
,
194111
(
2011
).
74.
F. J.
Dyson
, “
A Brownian-motion model for the eigenvalues of a random matrix
,”
J. Math. Phys.
3
,
1191
(
1962
).
75.
H.
Spohn
, “
Tracer dynamics in Dyson’s model of interacting Brownian particles
,”
J. Stat. Phys.
47
,
669
(
1987
).
76.
F.
Bouchet
and
T.
Dauxois
, “
Prediction of anomalous diffusion and algebraic relaxations for long-range interacting systems, using classical statistical mechanics
,”
Phys. Rev. E
72
,
045103
(
1992
).
77.
P. H.
Chavanis
,
C.
Rosier
, and
C.
Sire
, “
Thermodynamics of self-gravitating systems
,”
Phys. Rev. E
66
,
036105
(
2002
).
78.
C.
Sire
and
P. H.
Chavanis
, “
Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions
,”
Phys. Rev. E
66
,
046133
(
2002
).
79.
G. S.
Manning
, “
Limiting laws and counterion condensation in polyelectrolyte solutions I. Colligative properties
,”
J. Chem. Phys.
51
,
924
(
1969
).
80.
D.
Mondal
and
M.
Muthukumar
, “
Ratchet rectification effect on the translocation of a flexible polyelectrolyte chain
,”
J. Chem. Phys.
145
,
084906
(
2016
).
81.
E.
Levine
,
D.
Mukamel
, and
G. M.
Schütz
, “
Long-range attraction between probe particles mediated by a driven fluid
,”
Europhys. Lett.
70
,
565
(
2005
).
82.
O.
Hirschberg
,
D.
Mukamel
, and
G. M.
Schütz
, “
Approach to equilibrium of diffusion in a logarithmic potential
,”
Phys. Rev. E
84
,
041111
(
2011
).
83.
O.
Hirschberg
,
D.
Mukamel
, and
G. M.
Schütz
, “
Diffusion in a logarithmic potential: Scaling and selection in the approach to equilibrium
,”
J. Stat. Mech.: Theory Exp.
2012
,
P02001
.
84.
A.
Ryabov
,
E.
Berestneva
, and
V.
Holubec
, “
Brownian motion in time-dependent logarithmic potential: Exact results for dynamics and first-passage properties
,”
J. Chem. Phys.
143
,
114117
(
2015
).
85.
D. R.
Cox
and
H. D.
Miller
,
The Theory of Stochastic Processes
(
CRC Press
,
2001
), see Eq. (74) in Chap. 5.
86.
See https://dlmf.nist.gov/ for Digital Library of Mathematical Functions, National Institute of Standards and Technology (NIST), U.S. Department of Commerce.
87.
M.
Abramowitz
and
I. A.
Stegun
,
Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables
(
National Bureau of Standards
,
1970
), see Eq. 9.6.1 in p. 374.
88.
A.
Erdélyi
,
W.
Magnus
,
F.
Oberhettinger
, and
F. G.
Tricomi
,
Tables of Integral Transforms
, Volume I, California Institute of Technology, Bateman Manuscript Project (Based, in Part, on Notes Left by Late Prof. Harry Batemann) (
McGraw-Hill Book Company, Inc.
,
1954
), see Eq. (4) in p. 200.
89.
A.
Laforgia
and
P.
Natalini
, “
Some inequalities for modified Bessel functions
,”
J. Inequal. Appl.
2010
,
253035
.
90.
C. W.
Gardiner
,
Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences
(
Springer-Verlag
,
2003
).
91.
C.
Xavier
,
Fortran 77 and Numerical Methods
(
New Age International
,
1994
).
92.
I. S.
Gradshtein
,
I. M.
Ryzhik
,
D.
Zwillinger
, and
V.
Moll
,
Table of Integrals, Series, and Products
, 8th ed. (
Academic Press; Elsevier, Inc.
,
2014
), see Eq. 12 of Sec. 3.474 in p. 371.
You do not currently have access to this content.