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Thermodynamic Limits to Anomalous Diffusion

    Patrick Pietzonka

    • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

• Physics 14, 116

Thermodynamics units constraints on the time window over which anomalous diffusion can happen—a discovering which may be related to the research of diffusion-controlled mobile processes.


Figure 1: Diffusion is a commonplace course of that’s topic to thermodynamic legal guidelines. Hartich and Godec have now derived new thermodynamic constraints on techniques that exhibit anomalous kinds of diffusion.

Diffusion is a commonplace phenomenon: Put a droplet of ink right into a tank of water, and it’ll concurrently unfold out and develop into increasingly diluted (Fig. 1). This implies that the chance of discovering a given ink particle at some place evolves from a sharply peaked distribution to a really broad one (Fig. 2, left). Typically, the width of this distribution, quantified by the variance, will increase linearly with time, in keeping with an empirical legislation referred to as Fick’s second legislation of diffusion. But this legislation could be defied by techniques starting from the microscopic (comparable to organic cells) to the macroscopic (comparable to astrophysical plasmas), which exhibit anomalous diffusion processes the place the variance will increase nonlinearly with time [1]. New analysis by David Hartich and Aljaž Godec, on the Max Planck Institute for Biophysical Chemistry in Germany, reveals that the interval over which anomalous diffusion can happen is bounded and that these bounds have deep connections to the underlying thermodynamics of the diffusive system [2].

Hartich and Godec arrive at this end result by finding out anomalous diffusion via the lens of the thermodynamic uncertainty relation (TUR) [3]. This relation applies to stochastic techniques pushed by some drive (comparable to a mechanical drive, a chemical response, or a voltage) right into a stationary nonequilibrium state. Such techniques produce observable outputs that accumulate over time, known as “currents.” Examples of currents embrace the space traveled by a random walker or the variety of molecules produced in an enzymatic response. At a finite temperature, these currents are at all times fluctuating, and their relative uncertainty could be quantified when it comes to the variance divided by the squared imply worth of the present. The TUR states that this uncertainty multiplied by the power used for driving the system (measured in models of the thermal power okBT) is at all times 2. Thus, the TUR expresses a common trade-off between precision and energetic price: The present can solely be quantified with excessive precision in additional energetically pushed techniques.

The first paper conjecturing the TUR [4] (and the next paper that supplied its mathematical proof [5]) targeted on purposes to biomolecular processes. But the relation’s wider implications for statistical physics—for warmth engines or normal Langevin techniques, as an illustration—quickly grew to become clear. Researchers have since developed variations of the TUR that push the bounds of its applicability to quantum techniques and nonstationary states. Yet the implications of the TUR for anomalous diffusion have to this point remained unexplored.

The settings for which the TUR has to this point been studied contain currents whose imply and variance, to a tough approximation, each improve linearly with time. Because such techniques exhibit an ever-decreasing relative uncertainty (outlined because the variance divided by the squared imply), the TUR can solely be glad by an ever-increasing energetic price, yielding an uncertainty-energy product that varies little over time. One might marvel how anomalous diffusion, by which the variance will increase nonlinearly by definition, might ever match into this image. The easy however placing reply given by Hartich and Godec: It doesn’t—no less than not for arbitrary timescales. For the TUR to carry, anomalous diffusion can solely happen inside a restricted time window.

Figure 2: (Left) In a pushed diffusion course of, each the imply and the width 𝜎x(t) of the particle displacement improve linearly with time t (from gentle to darkish orange). (Right) For an everyday diffusion course of, the logarithmic plot of the variance (𝜎x2) over time can be near (however by no means beneath) a diagonal line, which bounds a “forbidden zone.” For subdiffusive and superdiffusive processes, the width spreads extra slowly or extra rapidly, respectively. The thermodynamic uncertainty relation (TUR) units temporal limits to such habits, expressed via a essential time t.(Left) In a pushed diffusion course of, each the imply and the width 𝜎x(t) of the particle displacement improve linearly with time t (from gentle to darkish orange). (Right) For an everyday diffusion course of, the logarithmic plot of the variance (𝜎x2) over t… Show more

The researchers analyze the 2 normal kinds of anomalous diffusion—subdiffusion and superdiffusion—for which the TUR yields two complementary bounds (Fig. 2, proper). In subdiffusion, which could be seen in, for instance, the diffusion of macromolecules in crowded intracellular environments, the variance will increase extra slowly than it does in regular diffusion. To perceive the implications of this slower improve with respect to the TUR, think about a 1D system of particles present process diffusion pushed by an exterior drive. If the particles can not move each other on the road, analysis has proven that the variance of their displacement will increase with the sq. root of the time, which is slower than the conventional linear improve. Because the imply displacement nonetheless will increase linearly, the relative uncertainty on this system decreases extra rapidly than normal. At the identical time, the energetic price of driving the diffusion additionally will increase solely linearly. Inevitably, the TUR can be violated past a sure essential time ( t), which could be estimated with a easy method. Hartich and Godec conclude that with a view to keep away from this violation, subdiffusion should come to an finish at this time limit on the newest. In the context of this 1D system, the purpose at which subdiffusion ends corresponds to the purpose at which the blocking of 1 particle by one other now not inhibits additional diffusion.

The researchers present that for superdiffusive transport, the alternative is true. This form of anomalous diffusion is exhibited by, for instance, biomolecules topic to lively transport inside cells, however it may be abstracted as a particle system by which the 1D atmosphere of the subdiffusive case is changed by a comb-like geometry: Particles diffuse alongside a central trunk or spine, however they could additionally unfold alongside dead-end facet branches. Here, the uncertainty will increase extra rapidly than normal. This speedy improve poses no drawback over lengthy timescales as a result of the product of the relative uncertainty and the driving power retains rising past 2. However, early on, it implies that the uncertainty would have been smaller than allowed by the TUR. Hence, the essential time t units a decrease certain on the onset of superdiffusion.

With these new bounds on subdiffusion and superdiffusion instances (Fig. 2, proper), Hartich and Godec convey the established area of anomalous diffusion into the identical thermodynamic framework that encompasses common diffusion. Just just like the TUR itself, their end result holds for any form of driving, nevertheless removed from thermal equilibrium. The end result can be easy, within the sense that one solely must know the energetic price of the driving drive with a view to formulate the constraints on the diffusion course of. Although these constraints will essentially be unfastened in lots of conditions, additional analysis will present whether or not further details about a system can be utilized to refine them. For occasion, it might be fascinating to see whether or not thermodynamics gives bounds not solely on the time window but additionally on the mathematical properties of the anomalous diffusion noticed inside this time window. If that is the case, the newfound hyperlink between thermodynamics and anomalous diffusion might result in a strong, joint theoretical basis. Such a theoretical foundation could possibly be related for purposes in mobile biophysics, the place a typical phenomenon is the transport of molecules in crowded environments topic to a restricted availability of chemical gas.


  1. R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: A fractional dynamics approach,” Phys. Rep. 339, 1 (2000).
  2. D. Hartich and A. Godec, “Thermodynamic uncertainty relation bounds the extent of anomalous diffusion,” Phys. Rev. Lett. 127, 080601 (2021).
  3. J. M. Horowitz and T. R. Gingrich, “Thermodynamic uncertainty relations constrain non-equilibrium fluctuations,” Nat. Phys. 16, 15 (2019).
  4. A. C. Barato and U. Seifert, “Thermodynamic uncertainty relation for biomolecular processes,” Phys. Rev. Lett. 114, 158101 (2015).
  5. T. R. Gingrich et al., “Dissipation bounds all steady-state current fluctuations,” Phys. Rev. Lett. 116, 120601 (2016).

About the Author

Image of Patrick Pietzonka

Patrick Pietzonka is a theoretical physicist working within the area of statistical physics. He did his undergraduate research and obtained his Ph.D. on the University of Stuttgart, Germany, with an experimental analysis keep on the University of Barcelona, Spain. In 2018, he moved on to the University of Cambridge, UK, the place he’s a postdoctoral researcher within the Department of Applied Mathematics and Theoretical Physics and a junior analysis fellow at Corpus Christi College. The focus of his analysis is on lively matter and stochastic thermodynamics.

Subject Areas

Statistical PhysicsNonlinear Dynamics

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