--- title: "Distinguishing active motion at the same variance" description: "Add a persistent active force, construct equal-variance observations and test an effective-temperature explanation using correlation and response." canonical_url: "https://sandboxphysics.com/stories/membrane-mechanics/active-fluctuations/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/membrane-mechanics/active-fluctuations/" last_modified: "2026-10-28" --- [Membrane mechanics](https://sandboxphysics.com/stories/membrane-mechanics/) L4 · Chapter 8 # Distinguishing active motion at the same variance Sandbox Physics · 2026-10-28 Large fluctuations can come from a hotter bath or a persistent active force. Their amplitudes can match exactly. Looking at how long motion remembers its past, then measuring the response to a small push, tests whether “simply hotter” explains the observations. ## Before you begin Thermal modes and variance from Chapter 4 and temporal correlation from Chapter 5. We study one stable linear mode with exponentially correlated forcing, without assuming a complete cytoskeletal model. - Specify colored active forcing independently of equilibrium thermal noise and match a passive reference variance. - Use correlation and a known-force response to test a constant effective temperature. ## How long does one push persist? Describe a real mode by displacement $x$, stiffness $K$ and friction $\zeta$. Elastic relaxation has rate $\lambda=K/\zeta$. Thermal noise comes from a bath of known temperature. Add a zero-mean active force with memory time $\tau_a$. An Ornstein–Uhlenbeck process generates this force by relaxing toward zero under random kicks. Its force variance $F_a^2$ and memory time are independent driving parameters. Specify them from a driving model or an independent measurement. $$ \zeta\dot x=-Kx+f_a+F_{\rm ext}+\xi_T,\qquad \langle\xi_T(t)\xi_T(0)\rangle=2\zeta k_BT\,\delta(t) $$ Every term on the right has force units. White-noise correlation uses a Dirac delta to represent no finite memory on the modeled time scale. $$ \langle f_a(t)f_a(0)\rangle=F_a^2e^{-|t|/\tau_a} $$ Active and thermal forcing are independent here. Spatial correlation, feedback and non-Gaussian forcing would require additional assumptions. Source: B. Loubet ## Match the variances exactly Set the stationary mean evolution of $x^2$ and $xf_a$ to zero. The resulting variance has a thermal part $k_BT/K$ plus an active part controlled by force strength and memory. At fixed force variance, sufficiently fast kicks cancel before the membrane can respond. Define a passive comparison with the same $K,\zeta$ and temperature $T_{\rm eff}=K\langle x^2\rangle/k_B$. It has exactly the same theoretical variance. Temporal correlation provides an additional test of this variance match. $$ \langle x^2\rangle=\frac{k_BT}{K}+\frac{F_a^2}{K(K+\zeta/\tau_a)} $$ This stationary result follows from the specified linear dynamics, with no equilibrium assumption for the active noise. ## A small push gives an independent observable Apply a small force $F$ at time zero and hold it. The mean displacement change divided by force is the step response $\chi(t)$. Zero-mean forcing drops out of the linear mean equation, so changing active variance leaves this deterministic response unchanged. At thermal equilibrium, correlation and response obey an exact relation. Calculate or measure response independently before comparing them. Generating a supposed active response by inserting active correlations into an equilibrium formula would make the proposed test circular. $$ \chi(t)=\frac{\Delta\langle x(t)\rangle}{F}=\frac{1-e^{-\lambda t}}{K} $$ This response comes from the mean dynamics with an applied force small enough to remain in the linear range. $$ \chi_{\rm eq}(t)=\frac{C(0)-C(t)}{k_BT},\qquad C(t)=\langle x(t)x(0)\rangle $$ This form assumes stationary thermal equilibrium and a coordinate conjugate to the applied force. Test active correlations against independently obtained response at multiple times. ## Worked example with equal variance and different memory Choose dimensionless units with $K=\zeta=k_BT=1$, $\tau_a=2$ and $F_a^2=1.5$. Thermal variance is one and active excess variance is one, for a total of two. A passive comparison at $k_BT_{\rm eff}=2$ also has variance two. Solving the two-variable linear correlation equations gives $C_{\rm active}(t)=2e^{-t/2}$ for these parameters. The hotter passive process has $C_{\rm hot}(t)=2e^{-t}$. At $t=1$ the values are $1.21$ and $0.736$. Equal fluctuation amplitude hides different memory. Both independent step responses are $1-e^{-1}\simeq0.632$. Treating the active process as equilibrium at temperature two would instead predict $(2-1.21)/2\simeq0.393$. The mismatch tests the constant-temperature explanation against a second observable. The Chapter 8 example has equal variance but different memory. The two lines are analytic model correlations in dimensionless units; response is calculated independently. ## Return to the active membrane Sciortino and colleagues encapsulated microtubules and molecular motors in vesicles to study traveling membrane deformation. Read spatial scales, time correlations and internal-driver images together. Large amplitude by itself does not uniquely specify a driving mechanism. The animated contour combines linear spherical modes; the correlation–response diagnostic selects one real mode. It does not solve a microtubule network or self-consistent area–tension feedback. The fixed-excess-area theory of Loubet and colleagues additionally solves an area condition for tension. Recalculating that branch requires solving the area condition after changing activity. Source: A. Sciortino et al. · B. Loubet ## Try the reasoning ### [Separate equal-variance processes](https://sandboxphysics.com/labs/membrane-fluctuations/) 1. Choose the matched passive and active comparison. Inspect their contours and predict whether amplitude alone distinguishes them. 2. Vary active memory and compare correlation decay with the independently calculated small-force response. 3. Test whether variance-derived effective temperature explains every time point. Identify an observation that would require a more detailed drive or feedback model. ## Exercises and explanations 1. Must equal active and hot-passive variances imply equal temporal correlation? Read the explanation No. Variance fixes correlation only at zero time separation. Persistent forcing can lengthen memory despite equal values at the origin. 2. Why does increased active strength leave mean step response unchanged here? Read the explanation Additional noise has zero mean, and stiffness and friction have no deformation feedback. The same deterministic coefficients control mean response. Activity that changes material properties or feedback requires a new test. 3. If correlation and response agree with one effective temperature at one time, is equilibrium established? Read the explanation No. Test multiple time scales, frequencies or conjugate variables and account for response uncertainty. This example excludes the specified constant-temperature model. Broader diagnosis requires additional observables and error analysis. [Previous: How composition and curvature reshape each other](https://sandboxphysics.com/stories/membrane-mechanics/curvature-and-composition/) [Next: Reading from a tube formula to an active-membrane paper](https://sandboxphysics.com/stories/membrane-mechanics/paper-reading/) ## Sources and calculated scope 1. [B. Loubet, U. Seifert and M. A. Lomholt, Effective tension and fluctuations in active membranes, arXiv:1112.2919v2; Physical Review E 85, 031913 (2012)](https://arxiv.org/html/1112.2919v2) Version 2, 14 December 2011: quadratic quasi-spherical modes and a direct active force with exponential temporal correlations. Our equal-variance response comparison is an explicitly specified finite linear model. We do not reproduce the paper’s self-consistent excess-area tension curves or permeation and curvature-force cases. 2. [A. Sciortino et al., Active membrane deformations of a minimal synthetic cell, Nature Physics 21, 799–807 (2025)](https://doi.org/10.1038/s41567-025-02839-3) Version of Record, 24 March 2025, Figures 2–4 and Data/Code Availability: spatial spectra, correlations and microtubule-driven deformation. The article is CC BY 4.0; raw data and analysis are deposited at doi:10.5281/zenodo.11351857. We provide a reading exercise, not a refit of these experiments or a simulation of their cytoskeleton.