Non-monotonic displacement distribution of active random walkers

Eial Teomy; Yael Roichman; Yair Shokef

REVIEW 3 major objections 4 minor 72 references

Non-monotonic displacement distribution of active random walkers

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that in a homogeneous system a short-time displacement distribution with more than one peak exists only when walkers have positive forward persistence, and derives the exact distribution for arbitrary temporal correlations.

desk verdict Genuine exact calculation, but the central 'peaks imply persistence' theorem is false as stated because the proof ignores the n=0 term and assumes unimodal step lengths. read the letter →

arxiv 1908.07242 v1 pith:DMT54U4Y submitted 2019-08-20 cond-mat.stat-mech

Eial Teomy , Yael Roichman , Yair Shokef This is my paper · ORCID

classification cond-mat.stat-mech PACS 05.40.Fb

keywords activerandomwalkpersistentdisplacementdistributionvanHovenon-monotonicitydirectionalcorrelationstemporalrun-and-tumblemotion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Active particles in a homogeneous fluid can still show a non-monotonic displacement distribution, a short-time probability curve with several humps, and this paper pins down exactly what causes those humps. For a minimal persistent random-walk model with arbitrary step timing, the authors prove that more than one peak can appear only if the walker has a positive tendency to repeat its previous step in the same direction. Temporal correlations between hops, by contrast, change only the size, shape, and decay of the peaks, not whether they exist. The practical payoff is a diagnostic: peaks in a measured short-time displacement distribution reveal directional persistence, while their spacing and heights report the step length and the timing statistics.

What carries the argument

The load-bearing object is a three-channel persistent random-walk model: at each hop the walker either repeats its previous hop exactly (forward, $\gamma_f$), retraces it (backward, $\gamma_b$), or takes a fresh step whose length comes from $s(\ell)$ in a random direction ($\gamma_0=1-\gamma_f-\gamma_b$). The derivation writes the Fourier transform $\tilde p_n(k)$ of the $n$-hop displacement as a matrix product of operators $M_0,M_f,M_b$ acting on the first-hop distribution, which yields exact closed forms for small $n$ and an exact convolution formula for the total PDF with a Gaussian thermal bath. For the no-peaks theorem the essential tool is an integral recursion, $p_{n+1}(r)=\frac{1}{2a}\int_{|r-a|}^{r+a}\frac{r'}{r}p_n(r')\,dr'$ for fixed step length $a$, which lets the authors prove by induction that every $p_n(r)$ with $\gamma_f=0$ is non-increasing in $r$; Gaussian noise can only smooth further, so the full $P_r$ has at most one peak.

What would settle it

Simulate the model with $\gamma_f=0$, $\gamma_b=0$, and a two-peaked step length $s(\ell)=\frac{1}{2}[\delta(\ell-a)+\delta(\ell-2a)]$ in a homogeneous three-dimensional system; if $P_r(r,\tau)$ has two local maxima for some small noise width $\alpha$, the theorem in its general form is false.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is a no-go result with a positive counterpart. For an isotropic active random walk whose hops are either forward repeats of the previous hop (probability $\gamma_f$), exact reversals (probability $\gamma_b$), or fresh random hops of length drawn from $s(\ell)$, the displacement probability density $P_r(r,\tau)$ after a short interval $\tau$ has at most one local maximum whenever $\gamma_f=0$, regardless of the timing correlations $q_n(\tau)$ and regardless of the step-length distribution. Therefore any displacement distribution with more than one peak implies $\gamma_f>0$, i.e. positive directional correlations. The same calculation gives an exact Fourier-space expression for $P_r$ for arbitrary $q_n$ and $s(\ell)$, and the numerical examples show that peak spacing is set by the most likely hop length, while the decay of peak heights with $r$ distinguishes Poissonian from correlated timing.

Load-bearing premise

The proof's load-bearing premise is that a fresh random hop has a single preferred length; if a walker instead hops with several distinct favored lengths, peaks could appear from the hop-length distribution alone even with no forward persistence, which would break the paper's broad claim.

Editorial extensions

If this is right

In a homogeneous active suspension, detecting two or more peaks in the short-time displacement distribution is direct evidence that the particles persist forward, and the short-time data can therefore be used to infer bounds on orientational correlations.
The spacing between peaks gives the most likely single-step length, while the relative heights of successive peaks give information about temporal correlations: exponential height decay signals Poissonian timing, and faster or slower decay signals negative or positive timing correlations.
Temporal correlations alone cannot produce non-monotonicity in a homogeneous system; they only set how pronounced the peaks are.
For heavy-tailed step lengths such as a Cauchy distribution the displacement PDF stays monotonic regardless of persistence, so the non-monotonicity diagnostic must be applied to systems with a well-defined single preferred hop length.
The exact formula for $P_r$ applies to any temporal correlation pattern $q_n$ and any step-length distribution in three dimensions, with analogous formulas in one and two dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension the paper leaves implicit is computing the minimal persistence $\gamma_f$ needed for a first side peak as a function of noise width $\alpha$ and step-length variance, which would turn the qualitative 'strong enough' condition into a quantitative threshold.
Because backward moves are shown to be equivalent to reweighting the temporal correlations, the no-peaks theorem implies that purely anti-persistent walkers ($\gamma_b>0$, $\gamma_f=0$) also cannot produce multiple peaks, so experimental anti-persistence should show a monotone displacement curve.
If the step-length distribution itself has several preferred lengths, the 'peaks imply persistence' slogan could fail; an experimenter using this diagnostic should first check that the single-step length distribution is unimodal.
The result suggests a two-point assay for living cells: measure the short-time displacement distribution in a homogeneous environment; if it is non-monotonic, the underlying motion has positive velocity autocorrelation even when long-time trajectories look diffusive.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [Name],

The short version: the exact displacement PDF for a persistent random walk with arbitrary temporal correlations is a real, useful result. The headline claim—that multiple peaks in the displacement distribution imply positive persistence—is not true as stated, and the proof has a concrete hole even within the authors' own model.

What is genuinely new: the operator recursion leading to Eq. (28) is systematic and checkable, the small-n expressions are correct in the cases I worked by hand, and the explicit evaluation for delta, modified Gaussian, and Cauchy step distributions with Gaussian noise is careful. The idea that the van Hove function's peak structure can be used to infer persistence is appealing and, under the right conditions, likely correct.

The problems are two. First, the proof of the no-multiple-peaks theorem in Section IV only treats p1 and p_n≥2; it never treats p0, the walkers that perform zero hops. With q0(τ)>0, as in their Poisson, binomial, and geometric models, the continuous Gaussian process turns p0 into a peak at r=0. With delta steps and small α, the q1 term is a shell peaked near r=a. So for γf=0 you get two local maxima, e.g., Poisson q0=e^{-τ}, q1=τe^{-τ}, a=1, α=0.1. The abstract's claim is not an overstatement; it is simply false as written. The second gap is the reduction to delta step sizes: smearing can only remove peaks if the step-length distribution is unimodal. If s(ℓ) has several preferred lengths, p1 already has several peaks at γf=0. That assumption is never stated.

There is also a semantic mismatch: the abstract says 'non-monotonicity', while the theorem actually concerns 'more than one peak'. A single peak at r=a already makes Pr non-monotonic, and that can happen at γf=0 (even with p0 ignored).

None of this kills the exact expression, which is worth having. But the necessary-condition theorem needs a qualified restatement: for a single-peaked step distribution and conditioning on at least one hop, at most one peak away from r=0 exists at γf=0. I would send this to a referee, because the main calculation is genuinely useful and the error is fixable with major revision. I would not cite it in its current form.

Best,

Referee Report

3 major / 4 minor

Summary. The paper studies a discrete-time random-walk model for active motion in d dimensions, in which each hop is either forward with probability γf, backward with probability γb, or uncorrelated (random length and direction) with probability γ0. The timing of hops is governed by arbitrary waiting-time distributions qn(τ), and a Gaussian continuous process is added by convolution. The authors derive a Fourier-space expression for the displacement distribution, give explicit formulas for small n and for three step-length distributions (Dirac, modified Gaussian, Cauchy), and present numerical plots for three temporal distributions (Poisson, binomial, geometric). The paper's central analytical claim is in Section IV: if γf=0, the displacement distribution has at most one peak, so the presence of more than one peak implies positive persistence. The abstract and summary generalize this to say that non-monotonicity of the displacement distribution occurs only when persistence is strong enough.

Significance. If the central theorem were valid, the paper would provide a clean and broadly applicable criterion: multiple peaks in a displacement distribution would diagnose forward persistence, independent of temporal correlations. The Fourier-space derivation is systematic and appears correct; the small-n formulas are explicit and checkable, and the Cauchy case is solved in closed form. The numerical exploration is useful and reproduces the expected qualitative dependence on step-size width and persistence. No fitted parameters are involved, and the model is simple enough that the main formulas could be rederived independently. However, the headline theorem as stated is not correct, and the abstract overstates what is proved. The paper's practical message can likely be repaired by restricting the statement, but the present version requires substantial revision of the central claim.

major comments (3)

[Section IV, Eqs. (30), (51), and (46)] The proof that γf=0 implies at most one peak omits the n=0 term. In Eq. (30), p0(r) is a delta function at r=0, and q0(τ)>0 for standard temporal distributions such as the Poisson distribution in Eq. (46), where q0=e^{-τ}. With s(l)=δ(l-a) and small α, the total distribution Pr(r,τ) contains a central peak near r=0 from the n=0 term and a shell peak near r=a from the n=1 term, giving two local maxima even when γf=γb=0. The induction argument for pn with n≥2 therefore does not establish at most one peak of the full sum. The theorem needs an additional condition, such as conditioning on at least one hop (or requiring q0=0), or an explicit exclusion of the r=0 feature.
[Section IV, third paragraph] The reduction to a Dirac delta step-size distribution is not valid for general s(l). Equation (30) gives p1(r)=s(r)/(Ω_d r^{d-1}), so if s(l) is multimodal, p1 already has several peaks at γf=0. The statement that any other step-size distribution only smears peaks presupposes that s(l) is single-peaked. As written, the paper claims the theorem for arbitrary step-length distributions, but the argument only covers single-peaked ones; the theorem and abstract must be restricted accordingly.
[Abstract and Section V] The paper conflates non-monotonicity with the presence of multiple peaks. A distribution with a single peak away from r=0 is non-monotonic, and the n=0 counterexample above shows non-monotonicity can occur without any persistence when both p0 and p1 contribute. What the (qualified) proof can support is, at most, a statement about the absence of multiple peaks under restricted conditions. The abstract's claim that 'non-monotonicity can occur only if the persistence is strong enough' is not established by the paper's own analysis and should be reformulated to match the actual theorem.

minor comments (4)

[Eq. (51)] In the expression for p1(r), the argument of the delta function should be r-a rather than l-a; as written the notation is inconsistent with the left-hand side depending on r.
[Figs. 3-5] The plots show Pr(r) on a log scale, but the text does not state clearly whether the plotted quantity is the vector density, the radial density, or some other marginal. This distinction is essential for interpreting 'peaks', especially for the delta-shell p1 contribution.
[References] Reference [29] cites Phys. Rev. Lett. volume 188, which does not exist; the volume should likely be 118. Reference [47] cites Phys. Rev. Lett. 36, 823 (1930); this should be Phys. Rev. 36, 823.
[Section II.C] The phrase 'It is a representative of general distributions with a single peak' should be explicitly connected to the restriction needed in Section IV; otherwise the reader may assume the theorem covers the Cauchy distribution and other heavy-tailed or multimodal step sizes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the peak condition is derived from the model recursion, not assumed or fitted.

full rationale

The paper constructs Pr(r,τ) from an explicit convolution of a discrete active process and an independent Gaussian thermal process, with the active process generated by the recursion relation (16). The characteristic functions p̃_n(k) in Eqs. (24)–(33) follow algebraically from that recursion and from the stated definitions of γ0, γf, and γb; no fitted constant is later relabeled as a prediction, and no benchmark or data is used to select the central result. The 'no peaks without orientational correlations' theorem is argued by an induction on p_n(r) with explicit p_2 and p_3 bases and derivative bounds in Section IV; whether that proof is fully correct is a mathematical question, not a circularity question. The self-citations in the reference list, such as Refs. [44] and [61], are background examples of related work and are not load-bearing for the derivation. Even if the theorem has a correctness gap involving the n=0 term or multimodal step-size distributions, as a skeptical reader might contend, that gap would be an overgeneralization or an error in the theorem's stated scope, not a reduction of the claimed result to its own inputs. The central derivation is self-contained and externally checkable against the recursion, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation introduces no new physical entities and fits no parameters to data. Illustrative plot values such as a=1, alpha=0.1, gamma_f values, and temporal distribution parameters are chosen by hand but do not enter the central theorem.

assumptions (5)

domain assumption The continuous thermal process is isotropic Gaussian with variance alpha^2 and is independent of the discrete active process (Eqs. 1-2).
Used to convolve discrete and continuous displacements; if the noise is non-Gaussian, the peak-smearing argument in Section IV could fail.
domain assumption Temporal correlations q_n(tau) are independent of hop lengths and directions, so the discrete PDF factorizes as the sum of q_n p_n (Eq. 5).
This decomposition is used throughout; if timing were correlated with step direction, temporal correlations could affect the existence of peaks.
ad hoc to paper For the no-peaks proof it suffices to consider Dirac delta step sizes, because any other step size distribution only smears peaks (Section IV, third paragraph).
This reduction is only valid for single-peaked step-length distributions; the model as stated in Section II allows arbitrary s(l), so the theorem's scope is narrower than claimed.
domain assumption The initial condition and environment are isotropic and homogeneous, so the displacement PDF depends only on r (Section II, assumptions 1-3).
Limits the result to homogeneous systems as stated in the abstract.
standard math Standard Fourier inversion, confluent hypergeometric identities, and integral representations are used without proof.
These are standard tools in the random walk literature and are not the source of the paper's claim.

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Cite this review

Pith. "Pith review of Non-monotonic displacement distribution of active random walkers." pith.science (2026). https://pith.science/paper/DMT54U4Y

@misc{pith2026190807242,
  author       = {Pith},
  title        = {Pith review of: Non-monotonic displacement distribution of active random walkers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMT54U4Y}},
  note         = {Machine review of arXiv:1908.07242}
}

read the original abstract

We consider a simple model for active random walk with general temporal correlations, and investigate the shape of the probability distribution function of the displacement during a short time interval. We find that under certain conditions the distribution is non-monotonic and we show analytically and numerically that the existence of the non-monotonicity is governed by the walker's tendency to move forward, while the correlations between the timing of its active motion control the magnitude and shape of the non-monotonicity. In particular, we find that in a homogeneous system such non-monotonicity can occur only if the persistence is strong enough.

Figures

Figures reproduced from arXiv: 1908.07242 by the authors.

**Figure 2.** FIG. 2: The step size distribution [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** FIG. 3: The PDF of the walker’s displacement [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: The PDF of the walker’s displacement [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: FIG. 5: The distribution [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

72 extracted references · 68 canonical work pages

[1]

de Groot and L

B. de Groot and L. Grubm¨ uller, Curr. Opin. Struct. Biol. 15, 176 (2005)

work page 2005
[2]

(36) For all three distributions the most likely step size is a

a modiﬁed Gaussian distribution sG (𝓁) = (𝓁 a )2ν e−ν𝓁2/a2 2νν+1/2 aΓ ( ν + 1 2 ), (35) which in the limit ν→∞ converges to the Dirac distribution, and 3) a Cauchy distribution sC (𝓁) = 4a𝓁2 π (𝓁2 +a2)2. (36) For all three distributions the most likely step size is a. The mean step size for the three distributions is ⟨𝓁⟩D = ∫ ∞ 0 𝓁sD (𝓁)d𝓁 =a, ⟨𝓁⟩G = ∫ ∞ ...

work page
[3]

Pr =Pr (r,τ ) depends only on the magnitude r =|⃗ r| of the displacement

the initial condition is isotropic, then when averaging over multiple particles their motion 3 is isotropic, i.e. Pr =Pr (r,τ ) depends only on the magnitude r =|⃗ r| of the displacement. The probability Pr (⃗ r,τ) is given by the following convolution of these two sources of ﬂuctuation Pr (⃗ r,τ) = ∫ Pd (⃗ r′,τ )Pc (⃗ r−⃗ r′,τ )d⃗ r′, (1) where Pd (⃗ r,τ...

work page
[4]

(48) In the Poissonian distribution there are no temporal correlations between the hops

binomial qbin n (τ) =  τ n  pn (1−p)τ−n, (47) with p = 0.6, and 3) geometric qgeo n (τ) =τ−1( 1−τ−1)n . (48) In the Poissonian distribution there are no temporal correlations between the hops. The main qualitative diﬀerence between the binomial and the geometric distribution is that for the geometric distribution ∂qgeo n ∂n < 0 always, i.e. more steps...

work page
[5]

H¨ oﬂing and T

F. H¨ oﬂing and T. Franosch, Rep. Prog. Phys.76, 046602 (2014)

work page 2014
[6]

Bechinger, R

C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Rev. Mod. Phys. 88, 045006 (2016)

2016
[7]

Metzler, J

R. Metzler, J. Jeon, and A. Cherstvy, BBA-Rev. Biomembranes 1858, 2451 (2016)

work page 2016
[8]

Hakim and P

V. Hakim and P. Silberzan, Rep. Prog. Phys. 80, 076601 (2017)

work page 2017

Show all 72 references

[9]

Norregaard, R

K. Norregaard, R. Metzler, C. Ritter, K. Berg-Sørensen, and L. Oddershede, Chem. Rev. 117, 4342 (2017)

2017
[10]

Hasnain and P

S. Hasnain and P. Bandyopadhyay, J. Chem. Phys. 143, 114104 (2015)

2015
[11]

Detcheverry, Europhys

F. Detcheverry, Europhys. Lett. 111, 60002 (2015)

2015
[12]

Rupprecht, O

J. Rupprecht, O. B´ enichou, and R. Voituriez, Phys. Rev. E 94, 012117 (2016)

2016
[13]

Detcheverry, Phys

F. Detcheverry, Phys. Rev. E 96, 012415 (2017)

2017
[14]

Sevilla, Phys

F. Sevilla, Phys. Rev. E 94, 062120 (2016)

2016
[15]

Ariel, A

G. Ariel, A. Be’er, and A. Reynolds, Phys. Rev. Lett. 118, 228102 (2017)

2017
[16]

Fedotov and N

S. Fedotov and N. Korabel, Phys. Rev. E 95, 030107 (2017)

2017
[17]

Wioland, E

H. Wioland, E. Lushi, and R. Goldstein, New J. Phys. 18, 075002 (2016)

2016
[18]

Stenhammer, C

J. Stenhammer, C. Nardini, R. Nash, D. Marenduzzo, and A. Mozorov, Phys. Rev. Lett. 119, 028005 (2017)

2017
[19]

Berthier and J

L. Berthier and J. Kurchan, Nat. Phys. 9, 310 (2013)

2013
[20]

Viscek, A

T. Viscek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995)

1995
[21]

Sep´ ulveda, L

N. Sep´ ulveda, L. Petitjean, O. Cochet, E. Grasland-Mongrain, P. Silberzan, and V. Hakim, PLoS Comput. Biol. 9, 1002944 (2013)

2013
[22]

Großmann, F

R. Großmann, F. Peruani, and M. B¨ ar, Phys. Rev. E 94, 050602 (2016)

2016
[23]

Liebchen and D

B. Liebchen and D. Levis, Phys. Rev. Lett, 119, 058002 (2017)

2017
[24]

Zimmerman, B

J. Zimmerman, B. Camley, W. Rappel, and H. Levine, Proc. Natl. Acad. Sci. USA 113, 2660 (2016). 23

2016
[25]

Farhadifar, J

R. Farhadifar, J. R¨ oper, B. Aigouy, S. Eaton, and F. J¨ ulicher, Curr. Biol.17, 2095 (2007)

2007
[26]

Staple, R

D. Staple, R. Farhadifar, J. R¨ oper, B. Aigouy, S. Eaton, and F. J¨ ulicher, Eur. Phys. J. E33, 117 (2010)

2010
[27]

S´ andor, A

C. S´ andor, A. Lib´ al, C. Reichhardt, and C. Reichhardt, Phys. Rev. E95, 032606 (2017)

2017
[28]

Reichhardt and C

C. Reichhardt and C. Reichhardt, Soft Matter 10, 7502 (2014)

2014
[29]

Zacherson, C

C. Zacherson, C. Wolﬀ, C. Whitchurch, and M. Toth, Phys. Rev. E 95, 012408 (2017)

2017
[30]

Reichhardt and C

C. Reichhardt and C. Reichhardt, Phys. Rev. E 90, 012701 (2014)

2014
[31]

K. Lam, M. Schindler, and O. Dauchot, New J. Phys. 17, 113056 (2015)

2015
[32]

Graf and E

I. Graf and E. Frey, Phys. Rev. Lett. 188, 128101 (2017)

2017
[33]

O. G. Illien P, B´ enichou O and V. R, J. Stat. Mech. 2015, P11016 (2015)

2015
[34]

Berg and L

H. Berg and L. Turner, Biophys. J. 58, 919 (1990)

1990
[35]

Ghosh, A

S. Ghosh, A. Cherstvy, and R. Metzler, Phys. Chem. Chem. Phys. 17, 1847 (2015)

2015
[36]

Schulz, A

J. Schulz, A. Kolomeisky, and E. Frey, Europhys. Lett. 95, 30004 (2011)

2011
[37]

Hermann, R

C. Hermann, R. Metzler, and R. Engbert, Sci. Rep. 7, 12958 (2017)

2017
[38]

Tchen, J

C. Tchen, J. Chem. Phys. 20, 214 (1952)

1952
[39]

Kareiva and N

P. Kareiva and N. Shigesada, Oeocologia 56, 234 (1983)

1983
[40]

Bogu˜ na, J

M. Bogu˜ na, J. Porr´ a, and J. Masoliver, Phys. Rev. E58, 6992 (1998)

1998
[41]

Romanczuk, M

P. Romanczuk, M. B¨ ar, W. Ebeling, B. Lindner, and L. Schminasky-Geier, Eur. Phys. J.202, 1 (2012)

2012
[42]

Ghosh, Y

P. Ghosh, Y. Li, G. Marchgiani, and F. Marchesoni, J. Chem. Phys. 143, 211101 (2015)

2015
[43]

Tahir-Kheli and R

R. Tahir-Kheli and R. Elliot, Phys. Rev. B 27, 844 (1983)

1983
[44]

Tahir-Kheli, Phys

R. Tahir-Kheli, Phys. Rev. B 27, 7229 (1983)

1983
[45]

Lettinga and E

M. Lettinga and E. Grelet, Phys. Rev. Lett. 99, 197802 (2007)

2007
[46]

Pouget, E

E. Pouget, E. Grelet, and M. Lettinga, Phys. Rev. E 84, 041704 (2011)

2011
[47]

Naderi, E

S. Naderi, E. Pouget, P. Ballesta, P. van der Schoot, M. Lettinga, and E. Grelet, Phys. Rev. Lett. 111, 037801 (2013)

2013
[48]

Sonn-Segev, A

A. Sonn-Segev, A. Bernheim-Groswasser, and Y. Roichman, Soft Matter 13, 7352 (2017)

2017
[49]

Goldstein, Q

S. Goldstein, Q. J. Mech. Appl. Math. 4, 129 (1951)

1951
[50]

Uhlenbeck and L

G. Uhlenbeck and L. Ornstein, Phys. Rev. Lett. 36, 823 (1930)

1930
[51]

Montroll and G

E. Montroll and G. Weiss, J. Math. Phys. 6, 167 (1965)

1965
[52]

Shlesinger, B

M. Shlesinger, B. West, and J. Klafter, Phys. Rev. Lett. 58, 1100 (1987). 24

1987
[53]

P. Levy, B. Soc. Math. Fr. 67, 1 (1939)

1939
[54]

Rayleigh, Philos

F. Rayleigh, Philos. Mag. 10, 73 (1880)
[55]

de Nigris, T

S. de Nigris, T. Carletti, and R. Lambiotte, Phys. Rev. E 95, 022113 (2017)

2017
[56]

Codling, M

E. Codling, M. Plank, and S. Benhamou, J. R. Soc. Interface 5, 813 (2008)

2008
[57]

Metzler, J

R. Metzler, J. Jeon, A. Cherstvy, and E. Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014)

2014
[58]

Sposini, A

V. Sposini, A. Chechkin, F. Seno, G. Pagnini, and R. Metzler, New J. Phys. 20, 043044 (2018)

2018
[59]

Jakub and R

S. Jakub and R. Metzler, New J. Phys. 20, 023026 (2018)

2018
[60]

Gnacik, A

M. Gnacik, A. Alsolami, and J. Burridge, J. Stat. Mech. p. 083207 (2018)

2018
[61]

Ben-Isaac, E

E. Ben-Isaac, E. Fodor, F. van Wijland, and N. Gov, Phys. Rev. E 92, 12716 (2015)

2015
[62]

Razin, R

N. Razin, R. Voituriez, and N. Gov, Phys. Rev. E 99, 022419 (2019)

2019
[63]

Magdziarz, Phys

M. Magdziarz, Phys. Rev. E 95, 022126 (2017)

2017
[64]

Ben-Isaac, Y

E. Ben-Isaac, Y. Park, G. Popescu, F. Brown, N. Gov, and Y. Shokef, Phys. Rev. Lett. 106, 238103 (2011)

2011
[65]

E. W. Weisstein, Conﬂuent Hypergeometric Function of the First Kind. From MathWorld—A Wolfram Web Resource , URL http://mathworld.wolfram.com/ ConfluentHypergeometricFunctionoftheFirstKind.html
[66]

Reynolds, L

A. Reynolds, L. Lepretre, and D. Bohan, Sci. Rep. 3, 3158 (2013)

2013
[67]

Palyulin, R

V. Palyulin, R. Mantsevich, R. Klages, R. Metzler, and V. Chechkin, Eur. Phys. J. B 90, 170 (2017)

2017
[68]

Johnson, J

D. Johnson, J. London, M. Lea, and J. Durban, Ecology 89, 1208 (2008)

2008
[69]

Kurihara, M

T. Kurihara, M. Aridome, H. Ayade, I. Zaid, and D. Mizuno, Phys. Rev. E 95, 030601 (2017)

2017
[70]

Mwaﬀo, R

V. Mwaﬀo, R. Anderson, S. Butail, and M. Porﬁrii, J. R. Soc. Interface 12, 20140884 (2015)

2015
[71]

E. W. Weisstein, Hermite Polynomial. From MathWorld—A Wolfram Web Resource , URL http://mathworld.wolfram.com/HermitePolynomial.html
[72]

E. W. Weisstein, Erf. From MathWorld—A Wolfram Web Resource , URL http:// mathworld.wolfram.com/Erf.html. 25

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

de Groot and L

B. de Groot and L. Grubm¨ uller, Curr. Opin. Struct. Biol. 15, 176 (2005)

work page 2005

[2] [2]

(36) For all three distributions the most likely step size is a

a modiﬁed Gaussian distribution sG (𝓁) = (𝓁 a )2ν e−ν𝓁2/a2 2νν+1/2 aΓ ( ν + 1 2 ), (35) which in the limit ν→∞ converges to the Dirac distribution, and 3) a Cauchy distribution sC (𝓁) = 4a𝓁2 π (𝓁2 +a2)2. (36) For all three distributions the most likely step size is a. The mean step size for the three distributions is ⟨𝓁⟩D = ∫ ∞ 0 𝓁sD (𝓁)d𝓁 =a, ⟨𝓁⟩G = ∫ ∞ ...

work page

[3] [3]

Pr =Pr (r,τ ) depends only on the magnitude r =|⃗ r| of the displacement

the initial condition is isotropic, then when averaging over multiple particles their motion 3 is isotropic, i.e. Pr =Pr (r,τ ) depends only on the magnitude r =|⃗ r| of the displacement. The probability Pr (⃗ r,τ) is given by the following convolution of these two sources of ﬂuctuation Pr (⃗ r,τ) = ∫ Pd (⃗ r′,τ )Pc (⃗ r−⃗ r′,τ )d⃗ r′, (1) where Pd (⃗ r,τ...

work page

[4] [4]

(48) In the Poissonian distribution there are no temporal correlations between the hops

binomial qbin n (τ) =  τ n  pn (1−p)τ−n, (47) with p = 0.6, and 3) geometric qgeo n (τ) =τ−1( 1−τ−1)n . (48) In the Poissonian distribution there are no temporal correlations between the hops. The main qualitative diﬀerence between the binomial and the geometric distribution is that for the geometric distribution ∂qgeo n ∂n < 0 always, i.e. more steps...

work page

[5] [5]

H¨ oﬂing and T

F. H¨ oﬂing and T. Franosch, Rep. Prog. Phys.76, 046602 (2014)

work page 2014

[6] [6]

Bechinger, R

C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Rev. Mod. Phys. 88, 045006 (2016)

2016

[7] [7]

Metzler, J

R. Metzler, J. Jeon, and A. Cherstvy, BBA-Rev. Biomembranes 1858, 2451 (2016)

work page 2016

[8] [8]

Hakim and P

V. Hakim and P. Silberzan, Rep. Prog. Phys. 80, 076601 (2017)

work page 2017