Extreme-Value Statistics of Stochastic Transport Processes: Applications to Molecular Motors and Sports

Alexandre Guillet; \'Edgar Rold\'an; Frank J\"ulicher

REVIEW 2 major objections 4 minor 94 references

Extreme-Value Statistics of Stochastic Transport Processes: Applications to Molecular Motors and Sports

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper shows that the complete finite-time statistics of the maximum and minimum of a biased random walk's position and entropy production are encoded in a single generating function, whose relaxation spectrum is the Marchenko-Pastur…

desk verdict The exact finite-time extrema statistics for the 1D biased random walk are the real prize here; the molecular-motor application rests on an under-validated effective reduction and the sports section is illustrative only. read the letter →

arxiv 1908.03499 v3 pith:6T5VMGTB submitted 2019-08-09 cond-mat.stat-mech cond-mat.softphysics.bio-ph

Alexandre Guillet , Édgar Roldán , Frank Jülicher This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.softphysics.bio-ph

keywords biasedrandomwalkextremevaluestatisticsentropyproductionfirst-passagetimesMarchenko-Pasturdistributionmatrixtheorymolecularmotorssportsscoring

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

The paper establishes that for a continuous-time biased random walk, the complete finite-time statistics of the maximum and minimum of displacement and entropy production are exactly computable from one generating function. Every moment and every extremal probability is fixed by two parameters, the bias $A$ and the jump rate $\nu$, with all time dependence carried by a relaxation spectrum that is exactly the Marchenko-Pastur law of random-matrix theory. The same formulas yield a symmetry between the distributions of minima and maxima of entropy production and a bound on how much entropy production can exceed its running average. A sympathetic reader cares because these quantities answer practical questions—how far a molecular motor backs up against its bias, or how long the weaker team can lead—and the paper gives closed-form answers rather than asymptotics alone.

What carries the argument

The central object is the generating function $G_{\min}(z;t)$ of the minimum distribution, expressed as an integral against the Marchenko-Pastur law—the limiting eigenvalue density of Wishart-Laguerre random matrices—on the relaxation-time interval $[\tau_0,\tau_\infty]$, with $\tau_0=(\sqrt{k_+}+\sqrt{k_-})^{-2}$ and $\tau_\infty=(\sqrt{k_+}-\sqrt{k_-})^{-2}$. The machinery is the Laplace transform of the first-passage-time density, $\hat P_{\rm fpt}(s;x)=e^{Ax/2}e^{-|x|\cosh^{-1}(s/2\nu+\cosh(A/2))}$, whose algebraic form turns the convolution over repeated first passages into the Marchenko-Pastur integral. The same first-passage duality $\tilde P_{\rm fpt}(T;x)=P_{\rm fpt}(T;-x)$ produces the max-min symmetry, and the effective-parameter formulas $\nu_{\rm eff}=\sum_\alpha\nu_\alpha$, $A_{\rm eff}=2\cosh^{-1}\left(\sum_\alpha\frac{\nu_\alpha}{\nu_{\rm eff}}\cosh\frac{A_\alpha}{2}\right)$ extend the 1D result to the 2D molecular-motor model.

What would settle it

Perform a high-precision simulation of a 1D biased random walk at, say, $A=1$, $\nu=1$, and compare the empirical distributions of $-S_{\min}(t)$ and $S_{\max}(t)-S(t)$ at $t=0.1, 1, 10$ against the paper's integral formula and the symmetry relation. Any statistically significant violation of the mirror symmetry, or a relaxation spectrum estimated from the time series that fails to match the Marchenko-Pastur density with parameter $\delta=e^{-A}$, would falsify the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that for a 1D continuous-time biased random walk with rates $k_\pm=\nu e^{\pm A/2}$, the generating function of the finite-time minimum is exactly $G_{\min}(z;t)=1+\frac{1-z}{e^A-1}\int_{\tau_0}^{\tau_\infty}\frac{1-e^{-t/\tau}}{1+f(z)\,\tau}\rho(\tau/\bar\tau)\,\frac{d\tau}{\bar\tau}$, with $f(z)=k_+(z-1)+k_-(z^{-1}-1)$ and $\rho$ the Marchenko-Pastur density. All moments and probabilities of the minimum, and by a mirror symmetry likewise of the maximum, follow by differentiating this expression. The paper proves $P(S(0)-S_{\min}(t)=s)=P(S_{\max}(t)-S(t)=s)$ and derives the supremal bound $\langle S_{\max}(t)\rangle-\langle S(t)\rangle\le 1$, generalizing the infimum law of stochastic thermodynamics. It further shows that the finite-time average minimum relaxes through the Marchenko-Pastur spectrum, so the extreme-value statistics of the walk can be estimated from the eigenvalues of a single Wishart or $\beta$-Laguerre random matrix.

Load-bearing premise

The motor and sports conclusions depend on compressing a genuinely multidimensional, time-varying process into a one-dimensional biased walk with fixed rates; the compression is exact for the 1D model, but for the molecular motor it is an unproven approximation checked at only one parameter set.

Editorial extensions

If this is right

Exact finite-time probabilities and moments for the extrema of any 1D biased random walk can be computed directly from the generating function, so no simulation is needed for the idealized model.
The distribution of the maximum of entropy production above its running value is the mirror image of the distribution of the minimum below its initial value at every time; measuring one gives the other.
A single sufficiently large Wishart or $\beta$-Laguerre random matrix with rectangularity $e^{-A}$ estimates the full time dependence of the average minimum to within a few percent in the tested cases.
For the two-dimensional motor model, the effective parameters of Eqs. (17)-(18) reproduce the simulated distributions of entropy extrema, and the average extreme displacements grow as the external force approaches the stall force while entropy extremes remain bounded.
For sports scores, the comeback window for the weaker team is bounded by $\tau_0\le t\le \tau_\infty$; the data suggest that in the 2018-19 Women's Champions League this window closes at roughly 60 minutes, while for Men's it extends past the 90-minute match.

Reading between the lines

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

The same Marchenko-Pastur structure should reappear for any Markov process whose Laplace-transformed first-passage density is an algebraic function of one master quantity, which would make extreme-value statistics of a broader class of currents estimable from random-matrix spectra.
The mirror symmetry could be tested experimentally as an extremal fluctuation theorem: recording only minima in single-molecule trajectories would predict the distribution of maxima above the running value, and deviations would flag non-Markovian effects.
The sports analysis is testable with play-by-play score data: if real scoring rates are time-dependent, the empirical extrema should deviate systematically from the Poisson fixed-rate prediction, and the deviation would quantify the minimal model's limits.
Near stall, the predicted divergence of average extreme spatial steps could be probed in single-motor assays, since the divergence timescale depends on $A_{\rm eff}$ in a specific way and load-dependent measurements would discriminate the effective 1D mapping from alternative multi-state descriptions.

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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

The core of this paper is the exact finite-time generating function for the extrema of the 1D biased random walk, together with the Marchenko-Pastur relaxation spectrum and the max-min symmetry. Those are new, cleanly derived, and internally consistent. I checked the t→0 and t→∞ limits of Eq. (12), the change of variables to the MP density, and the duality argument behind Eq. (11) — all correct. The Gillespie simulations in Figs. 2–3 and 5–6 match the formulas, which is gratifying. The random-matrix estimators (Wishart and Laguerre) are a clever practical device and the convergence discussion in Appendix D is honest about the bias.

The soft spots are all at the application layer, and they are real but not fatal. The effective-1D reduction for the molecular motor (Eqs. 17–18) matches only the total jump rate. It does not match the mean entropy current or the entropy diffusivity of the 2D model, so there is no reason it should be exact for generic parameters. The authors know this—they call it an approximation—but the abstract says the approach “captures key features” of motor extreme events, and that claim is supported by a single parameter set. That needs either a derivation from the 2D master equation or a systematic scan over a range of affinities and rates. The sports section is explicitly illustrative and less load-bearing; still, the prescriptive comment about shortening women's matches overreaches, and the text in Sec. V muddles τ0 with the mean waiting time τ1. The global-infimum result is recycled from the authors' own Ref. [31], but that is properly cited and it is a component, not the contribution.

Who gets value: stochastic thermodynamics researchers who want exact finite-time extrema statistics for biased random walks, and anyone interested in the random-matrix connection. The motor and sports applications should be read as examples, not as demonstrated universal claims. The paper deserves a serious referee — the maths is solid and the central result is publishable — but the motor-reduction claim needs to be tempered or strengthened before acceptance. I would send it to peer review, expecting major revision on the application sections.

Referee Report

2 major / 4 minor

Summary. The manuscript derives exact finite-time statistics of the extrema (maximum and minimum) of the position and of the entropy production of a one-dimensional continuous-time biased random walk. Starting from the first-passage-time distribution, the authors obtain generating functions and integral representations for the average extrema (Eqs. (9), (12), (14)), show that the relaxation spectrum is the Marchenko-Pastur law, and use this to propose Wishart/Laguerre random-matrix estimators. They then apply the 1D formulas to a two-dimensional stochastic model of a molecular motor through effective parameters (Eqs. (17)-(18)) and illustrate the relaxation-time spectra with scoring data from soccer, handball, and basketball.

Significance. If the 1D derivation is correct, this is a valuable exact contribution: it generalizes the infimum law for entropy production, establishes a symmetry between the distributions of minima and maxima (Eq. (11)), provides a finite-time supremum bound (Eq. (21)), and connects finite-time extreme-value statistics to random-matrix theory. The 1D derivations are largely self-contained and internally consistent; I verified the t->0 and t->infinity limits of Eq. (12), the change of variables leading to the Marchenko-Pastur integral in Appendix C, and the duality argument behind Eq. (11). The Gillespie simulations in Figs. 2 and 3 support the 1D results. The main weakness is the transfer to the 2D molecular-motor model: the effective-1D reduction is heuristic and is validated for a single parameter set, so the abstract-level claim that the approach captures key features of motor extreme events is not yet established. The sports section is explicitly illustrative and is not the main scientific claim.

major comments (2)

[IV, Eqs. (17)-(18)] The reduction of the 2D molecular-motor model to an effective 1D biased walk is not derived from the 2D master equation. In the 2D model S(t)=A_m X(t)+A_c Y(t) changes by four step affinities A_m, A_c, A_m+A_c, and A_c-A_m, with mean entropy current sigma_2D = sum_alpha 2 nu_alpha A_alpha sinh(A_alpha/2) and entropy diffusivity D_2D = sum_alpha nu_alpha A_alpha^2 cosh(A_alpha/2). Equations (17)-(18) match only the total jump rate k_eff^+ + k_eff^- = sum_alpha (k_alpha^+ + k_alpha^-); they impose no matching of sigma_2D or D_2D. Since finite-time extrema are controlled by both drift and fluctuations, the effective model cannot be exact for generic parameters. The validation in Fig. 5 is a single Gillespie parameter set, and no comparison of sigma_eff and D_eff with sigma_2D and D_2D is reported. I ask the authors to derive or justify the reduction, to report these cumulant comparisons for the parameters used, and to test the approximation over a range of affinities; without this, the abstract's claim about molecular-motor extreme events is not supported.
[IV, Fig. 5(c)] The distribution comparison in Fig. 5(c) uses Eq. (3), which is the infinite-time global minimum distribution, to validate simulation data at t=50 ms. The paper's central exact result is the finite-time statistics encoded in Eqs. (5) and (9); Fig. 5(b) tests only the average finite-time behavior, while Fig. 5(c) tests only the long-time limit. Please compare the motor simulations to the finite-time distribution (5) or the corresponding generating function, and state the relevant relaxation times for the motor parameters so the reader can judge whether t=50 ms is indeed in the asymptotic regime. As written, the finite-time distribution claim for the motor model is not validated.

minor comments (4)

[Fig. 4 caption] The caption states that a single 64x64 random matrix is used, while the legend and text report m=22, 24, and 26; please reconcile these numbers.
[Appendix E, Eq. (E5)] The notation for the Kampe de Feriet function in Eq. (E5), written as '2+0F1+1', is very hard to parse; please give the explicit parameter vectors or a reference with the exact convention used.
[Fig. 2 inset] The inset shows curves for 0.5 <= A <= 5 with nu=1, but there is no legend identifying which curve corresponds to which A; add a legend or a color scale.
[Sec. V] The sentence 'For women's soccer matches one that therefore does not expect a comeback' contains a typo ('one that' should be 'one does'), and 'the weaker time falls behind' should read 'the weaker team'; also, the sports conclusions should explicitly remind the reader that the model assumes fixed Poisson scoring rates with no correlations or non-stationarity, as this assumption is not tested against within-match data.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the exact 1D extremal statistics are derived from the master equation; the Marchenko-Pastur representation is an explicit change of variables, and the motor application is an underived but non-circular effective approximation.

full rationale

The central derivation is self-contained. The finite-time extrema statistics of the 1D biased random walk follow from the first-passage-time density (Eq. A19), obtained by solving the master equation via renewal and Laplace transforms, and the integral representation (Eq. B16) is an exact manipulation of Bessel-function identities. No parameter is fitted to the predicted quantity: A and nu are model inputs, and all moments and probabilities of the extrema are computed from them. The Marchenko-Pastur representation (Eqs. C6-C7) is an explicit change of variables in the exact integral, so the random-matrix estimators (Eqs. D2-D3) are numerical quadrature of the same exact result, not independently fitted predictions. The citations to Refs. [31] and [84] supply standard martingale absorption facts that are re-derived in Appendix A and are not used to define the finite-time extrema being predicted. The motor section's effective-1D reduction (Eqs. 17-18) is a heuristic approximation validated by Gillespie simulations, but it is not circular: it does not define the 2D result as the 1D answer by construction, and the exact 1D formulas remain independently derived. The sports application is an illustrative Poisson model with rates estimated from data, which is legitimate calibration rather than circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The exact 1D core is essentially parameter-free: A and nu are the model inputs, and nothing is fitted to the extrema data. The free parameters appear only in the sports illustration (k+ and k- fitted to aggregate goals). The load-bearing extra assumptions are the effective-1D reduction of the 2D motor model, which is heuristic and validated for one parameter set, and the fixed-rate Poisson sports model; both are acknowledged in the text but bound the scope of the applications.

free parameters (2)

k+, stronger-team scoring rate (sports application) = women: 0.042 +/- 0.006 goals/min; men: 0.025 +/- 0.010 goals/min
Estimated from aggregate goals scored by group-stage leaders in the 2018-2019 UEFA Champions League (Appendix F, Table I); analogous fits for handball and basketball appear in Fig. 8. All sports extreme-value timescales derive from these fits under a fixed-rate Poisson model.
k-, weaker-team scoring rate (sports application) = women: 0.006 +/- 0.002 goals/min; men: 0.009 +/- 0.002 goals/min
Estimated from aggregate goals conceded by group-stage leaders. The comeback timescales tau0, tau1, tau_bar and tau_infinity in Figs. 7-8 are functions of k+ and k- under a fixed-rate Poisson model, so all sports conclusions inherit these fitted values.

assumptions (6)

domain assumption Transition rates factor as k(x,y) = nu(x,y) exp[A(x,y)/2] with A antisymmetric, identifying A with per-jump entropy production (Eq. 1).
Physical input from local detailed balance; required for the interpretation S(t) = AX(t) used throughout the paper.
standard math First-passage-time Laplace transform to site -x equals the x-th power of the transform to site -1 (Eq. B7).
Exact for nearest-neighbor continuous-time walks (skip-free property); closes the generating functions in Eqs. (B8-B9).
standard math Empirical spectral density of Wishart and Laguerre-L matrices converges to the Marchenko-Pastur law with parameter delta = e^{-A} (Appendix D).
Marchenko-Pastur theorem (Ref. [59]) and Dumitriu-Edelman beta ensembles (Ref. [63]); the matrices are intentionally constructed with this delta, so the estimators converge to the exact integral by construction.
ad hoc to paper A 2D molecular motor can be reduced to an effective 1D biased walk with parameters (17)-(18).
Heuristic mapping based on matching total forward and backward rate sums; not derived from the 2D master equation and validated numerically only for the parameter set of Fig. 5.
ad hoc to paper Sports score differences are modeled as biased random walks with fixed Poisson scoring rates k+ and k- (Sec. V).
Acknowledged in the text as ignoring correlations and non-stationary effects; every sports conclusion, including the match-duration recommendation, inherits this assumption.
standard math e^{-S(t)} is a martingale and integral fluctuation relations hold at stopping times (Appendix A3).
Proved in the appendix via the Skellam generating function; used to obtain the absorption probability (A13) and the global minimum distribution (3).

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Pith. "Pith review of Extreme-Value Statistics of Stochastic Transport Processes: Applications to Molecular Motors and Sports." pith.science (2026). https://pith.science/paper/6T5VMGTB

@misc{pith2026190803499,
  author       = {Pith},
  title        = {Pith review of: Extreme-Value Statistics of Stochastic Transport Processes: Applications to Molecular Motors and Sports},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6T5VMGTB}},
  note         = {Machine review of arXiv:1908.03499}
}

read the original abstract

We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks. Our approach captures key features of extreme events in molecular motor motion along linear filaments. Our results generalize the infimum law for entropy production and reveal a symmetry of the distribution of its maxima and minima. We also show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Mar{\v{c}}enko-Pastur distribution of random-matrix theory. Using this result, we obtain estimates for the extreme-value statistics of stochastic transport from the eigenvalue distributions of suitable Wishart and Laguerre random matrices. We confirm our results by numerical simulations of stochastic models of molecular motors and discuss as illustrative example our theory in the context of sports.

Figures

Figures reproduced from arXiv: 1908.03499 by the authors.

**Figure 2.** Average minimum hSmin(t)i (blue open symbols) and average of the maximum minus the final value hSmax(t)i − hS(t)i (red filled symbols) of stochastic entropy production as a function of time of a 1D biased random walk. The symbols are averages over 10 sets of 103 numerical simulations; the error bars are the standard deviation of the mean values obtained from these sets. The black lines are obtained from numerical … view at source ↗

**Figure 3.** Empirical probability density of −Smin(t) (blue open symbols) and Smax(t) − S(t) (red filled symbols) obtained from 108 numerical simulations of a 1D biased random walk with parameters A = 1 and ν = 1. Different symbols represent different integration times t = 10−2 (squares), t = 10−1 (circles), t = 1 (up triangles), t = 10 (down triangles). The black lines are the theoretical distributions for different values of… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Finite-time average minimum of entropy production [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: Entropy production extrema for a two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: Numerical results of mechanical and chemical cur [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: Marˇcenko-Pastur density ρ(τ /τ¯)/τ¯ as a function of the relaxation time τ for the extreme-value statistics corresponding to a 1D biased random walk, where ¯τ is a characteristic timescale. The distributions are estimated for scores of women’s soccer (red line) and …

**Figure 8.** Figure 8: Marˇcenko-Pastur density ρ(τ /τ¯)/τ¯ as a function of the normalized relaxation time τ /τmatch for the extreme-value statistics corresponding to a 1D biased random walk, estimated for different sports. Here τmatch is the game duration of the respective sports. Paramet…

**Figure 9.** Figure 9: Numerical results for the random-matrix estimates [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

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Reference graph

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} denotes the position of the walker at time t≥ 0

Model and solution of the Master equation We consider a continuous-time biased random walk in a discrete one-dimensional lattice, where X(t) = {0,±1,±2,... } denotes the position of the walker at time t≥ 0. We assume that X(0) = 0 and that the walker can jump from state X(t) = x to x + 1 (x− 1) at a rate k+ (k−). The waiting time at any site is exponentia...
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Reviewed August 14, 2026 · model on record in the stance chip above.

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[2] [1]

} denotes the position of the walker at time t≥ 0

Model and solution of the Master equation We consider a continuous-time biased random walk in a discrete one-dimensional lattice, where X(t) = {0,±1,±2,... } denotes the position of the walker at time t≥ 0. We assume that X(0) = 0 and that the walker can jump from state X(t) = x to x + 1 (x− 1) at a rate k+ (k−). The waiting time at any site is exponentia...

[3] [2]

For this purpose we consider the scaling limit of Px(t) given by Eq

Large deviation and diﬀusion limit We now discuss and review large deviation properties of the 1D biased random walk [80, 81] and relate them to the statistics of 1D drift diﬀusion process. For this purpose we consider the scaling limit of Px(t) given by Eq. (A2) for largex∼vt, withv =k+−k− the net veloc- ity of the walker. We assume a large deviation pri...

[4] [3]

We ﬁrst show explicitly thate−S(t) is a martingale pro- cess with respect to X(t), i.e

Martingales and absorption probabilities In this subsection we employ martingale theory to de- rive an analytical expression of the absorption probability Pabs(−x) for a 1D biased random walk starting at x = 0 to ever reach an absorbing boundary located at −x< 0. We ﬁrst show explicitly thate−S(t) is a martingale pro- cess with respect to X(t), i.e. ⟨e−S(...

[5] [4]

It can also be derived from the distribu- tion of the walker using Laplace transforms through the renewal equation: Px(t) = ∫ t 0 Pfpt(t′;x)P0(t−t′)dt′

First-passage-time distribution The ﬁrst-passage-time density Pfpt(t′;x) can be de- rived from the solution of the Master equation (A1) with an absorbing boundary at site x⁄= 0, with x an integer number [54]. It can also be derived from the distribu- tion of the walker using Laplace transforms through the renewal equation: Px(t) = ∫ t 0 Pfpt(t′;x)P0(t−t′)...

[6] [5]

First we use the fact the the ﬁrst-passage-time density is the derivative of the absorption probability, ˆPfpt(s;x) = s ˆPabs(x;s)−δx,0, which holds for anyx

Finite time statistics in the Laplace domain We rewrite the probability-generating function Gmin using the ﬁrst-passage-time density derived above (A18). First we use the fact the the ﬁrst-passage-time density is the derivative of the absorption probability, ˆPfpt(s;x) = s ˆPabs(x;s)−δx,0, which holds for anyx. Equation (A19) implies that the Laplace tran...

[7] [6]

Integral representations of extreme value statistics We start from the ﬁrst-passage-time density for- mula (A19) and we exploit two properties of the mod- iﬁed Bessel function of the ﬁrst kind. This allows us to rewrite the absorption probability as a deﬁnite integral of trigonometric and hyperbolic functions and the parame- ters A and ν: Pabs(x;t) = ∫ t ...

[8] [7]

Average value of the ﬁnite-time minimum of entropy production Performing the changes of variablek = 2ν(cosh(A/2)− y), as well as τ = 1/k, in Eq. (B16) we obtain ⟨Smin(t)⟩ =−A 2π ∫ k∞ k0 1−e−kt k √ (k∞−k)(k−k0)dk k (C2) =−A 2π ∫ τ∞ τ0 ( 1−e−t/τ )√ (τ∞−τ)(τ−τ0) τ∞τ0 dτ τ (C3) where we have introduced the variables k∞/0≡ ( √ k+± √ k−)2 , (C4) τ∞/0≡ 1 k0/∞ . ...