### Brownian motion and the Langevin equation Einstein's *annus mirabilis* (1905) was famous for the publication of his papers on the special theory of relativity and the discovery of the photoelectric effect. But in that same year, he also published a foundational paper on the cause of Brownian motion. At the time, the molecular nature of fluids was still an unsettled question, and he used statistical mechanics arguments to establish that well-known properties of diffusion must be due to collisions between microparticles and even tinier fluid molecules in their environment. Soon thereafter, the French physicist Paul Langevin proposed a model for individual particle motion that was based on Newton's second law of motion. In 1908 he postulated (arguably) the first *stochastic differential equation*. Let $\{X(t), V(t)\}_{t \geq 0}$ be the time-dependent evolution of a particle's position and velocity. Then, written in modern notation, $ \begin{aligned} \dot X(t) &= V(t), \\ m \mathrm{d} V(t) &= -\gamma V(t) \mathrm{d} t + \sqrt{2 k_B T \gamma} \mathrm{d} W(t), \end{aligned} $ where $-\gamma V(t)$ is the drag force experienced by the particle, $k_B$ is Boltzmann's constant, and $T$ is the temperature of the fluid medium. The process $W(t)$ is Brownian motion, which can be thought of as a random walk in which changes of direction happen at every moment in time. (More on that apparent impossibility in a moment.) The first of Langevin's equations expresses the undergraduate calculus statement that the derivative of position is its velocity. The second equation states that the particle's acceleration arises from a balance of drag force and thermal fluctuations. It is no accident that the notation for derivatives is different in the two equations. In this system, the derivative of the position is the regular derivative from calculus that we all know and love. But the notation $\mathrm{d} V(t)$ and $\mathrm{d} W(t)$ is a confession that these are *not* derivatives in a traditional sense. In fact, as developed in a theory by N Wiener in the 1920s and K Itô in the 1940s, Brownian motion and the associated velocity process are so rough that they are, in fact, *nowhere differentiable*. What is written as a stochastic differential equation is just a convenient shorthand for the Langevin equation's true form, which is as a *stochastic integral equation*: $ V(t) - V_0 = -\frac{\gamma}{m} \int_0^t V(s) \mathrm{d} s + \frac{\sqrt{2 k_B T \gamma}}{m} W(t). $ The establishment of a mathematically sound version of these equations opens up a next generation of investigation in which reliable computational techniques can be established and generalizations can be articulated when particles interact with each other and with forces external to the local environment. #### Viscoelastic diffusion and the generalized Langevin equation *Funded 2014-2017 through NSF-Applied Math. PI with Christel Hohenegger (Utah).* In contrast to viscous fluids, viscoelastic fluids (like mucus and cytoplasm), have the ability to briefly store energy before reacting to force impulses. This delayed response is summarized by a memory kernel $K(t)$ and, for example, the drag force experienced by a particle at a time $t$ is given by an integral over all previously experienced drag forces, weighted by the memory kernel. When the drag term in the Langevin equation is changed to include memory, it turns out that the thermal fluctuation term needs to be modified as well. Since the agents that cause drag are the same as those that generate random collisions due to thermal fluctuations, Kubo and others developed the so-called *fluctuation-dissipation relationship*[^kubo]. In the end, we can write equations like $ m \dot{V}(t) = - \int_{-\infty}^t K(t-s) V(s) \mathrm{d} s + \sqrt{k_B T} F(t) $ where $F(t)$ is a stationary Gaussian process satisfying $\mathbb{E}\big(F(t) F(s)\big) = K(t-s)$. The role of the applied mathematician emerges here. Note the change back to conventional derivative notation on the left-hand side. Note the integral stretches back to negative infinity. Note that the thermal fluctuation term is no longer expressed as a derivative, and is now referred to as a colored noise instead of white noise. Note even that there is no longer a 2 in front of the $k_B T$. These are all mathematical claims, and their veracity is not at all obvious. Although physicists have been using versions of this generalized Langevin equation (GLE) since the 1960s, and its usefulness in microparticle movement modeling has been well-established since 1995[^mason-weitz], each of the aforementioned notes required rigorous mathematical investigation and clarification, so that we can then turn around and develop models for interacting particles and fluid-particle coupling. My signature mathematical work has been to address these issues in a way that is rigorous for the mathematics community, meaningful for the physics community, and practical for engineers and experimentalists. >**Featured paper:** **SA McKinley** and [HD Nguyen](https://sites.google.com/view/hungdnguyen), *[SIAM Journal on Mathematical Analysis](https://epubs.siam.org/doi/abs/10.1137/17M115517X)*, 2018} >In this work, my former PhD student and I established a near-complete mathematical theory for stationary solutions to the linear GLE, including well-posedness and regularity (whether or not legitimate derivatives exist). We also examined a long-standing observation in the physics literature theorem concerning the relationship between the memory kernel $K$ and the long-term behavior of GLE solutions[^morgado]. The existing physics methods either relied on special constructions of the memory kernel[^kupferman] [^kou] or used a non-stationary version of the GLE[^kneller]. Our approach was a direct analysis of the stationary GLE that addressed a much larger class of memory kernels. The main technical innovation was the extension of Abelian and Tauberian theorems from Laplace to Fourier transforms. This is because the large-time behavior of solutions is dictated by the small-frequency behavior of their Fourier transforms. | Diffusion in Viscous and Viscoelastic Media | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | ![[sucrose_lung.png]] | | Figure generated from data reported in Hill et al showing so-called *pathwise MSD curves* for populations of particles in mucus and in sucrose. This observation, along with a study of the particles' autocorrelation functions were the basis for our development of the GLE as our primary model for viscoelastic diffusion. | In parallel work with [Christel Hohenegger](https://sites.google.com/gcloud.utah.edu/christel-hohenegger/homepage), we established theory and computational methods for simulating a single particle immersed in a viscoelastic fluid through what is called the stochastic immersed boundary method. This work is an extension of a popular method used to simulate particles interacting through viscous fluids. In Hohenegger & McKinley (2017)[^hohenegger2017] we showed that passively moving particles inherit the memory structure of the fluid in which they are immersed in a predictable way. In our follow-up effort, Hohenegger & McKinley (2018)[^hohenegger2018], we investigated the standard method that engineers use to infer fluid properties from particle statistics. >**Featured paper:** C Hohenegger and **SA McKinley**, *SIAM Journal on Applied Mathematics*, 2018[^hohenegger2018] Rheology is the study of the flow and deformation of fluids. While classical rheology experiments involve applying stress directly to a fluid in order to measure its response, biological fluids like mucus and cytoplasm are not available in large enough quantities for this kind of study. The field of passive microrheology was born with the seminal paper by Mason \& Weitz[^mason-weitz], in which the authors demonstrated a technique for learning fluid properties from the movement statistics of individual immersed microparticles. In this paper, Christel Hohenegger and I used our new theoretical insights to revisit the Mason \& Weitz protocol along with its extensions and revisions[^squires]. We discovered something that has become a theme in my work: that precise inference of model parameters may not always be possible due to structural unidentifiability, but nevertheless the quantities of interest to experimentalists (in this case, the storarge and loss moduli of the fluid medium) are possible to estimate, and with great accuracy. In the course of writing this paper, Christel and I noticed that a commonly held assumption (that the velocity of the particle and the stochastic forcing are independent) was not true for the version of the GLE we studied. This inspired mathematical physicist Martin Hanke to take a closer look, and he recently posted this fascinating exploration of the "second fluctuation-dissipation theorem"[^hanke]. The aforementioned work concerns individual freely diffusing particles, but the behavior of subdiffusive particles being subjected to external forces, say optical traps[^goychuk2009] or molecular motor proteins bound to a microtubule[^goychuk2014][^bouzat] poses continuing theoretical and modeling challenges. In joint work with Hung Nguyen, David Herzog (Iowa) and former Tulane colleague Nathan Glatt-Holtz, we addressed an apparent paradox in which a subdiffusive particle can have long-term memory and yet exhibit a property called geometric ergodicity that allows systems to stabilize somewhat rapidly[^gh-nonlinearity]. Hung's work with another Tulane colleague, Gustavo Didier, has also addressed statistical challenges and core physics issues like equipartition of energy for subdiffusive particles subject to external forces. Still there remain both theoretical and modeling issues that stand as a barrier to simulating hydrodynamic interations among multiple particles in viscoelastic media. In a recent collaboration with current Tulane PhD student Irene Erazo-Estrada and colleagues Lisa Fauci and Ricardo Cortez, we are taking a first step in this direction by proposing an alternative framework to the immersed boundary method. By developing a computationally efficient external forcing model, we are modeling the fluctuating hydrodynamics of multiple interacting rigid spheres that preserves important properties like the Stokes-Einstein relationship between particle size and associated diffusivity. [^kubo]: R Kubo. [The fluctuation-dissipation theorem.](https://iopscience.iop.org/article/10.1088/0034-4885/29/1/306) *Reports on Progress in Physics* 29 (1):255, 1966. DOI: https://doi.org/10.1088/0034-4885/29/1/306. [^hohenegger2017]: Christel Hohenegger and **Scott A McKinley**. [Fluid–particle dynamics for passive tracers advected by a thermally fluctuating viscoelastic medium.](https://doi.org/10.1016/j.jcp.2017.03.053) *Journal of Computational Physics*, 340:688–711, 2017 [^mason-weitz]: TG Mason and DA Weitz. [Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.74.1250). Phys Rev Lett 74, 1250 DOI: https://doi.org/10.1103/PhysRevLett.74.1250 [^squires]: Todd M Squires and Thomas G Mason. [Fluid mechanics of microrheology](https://doi.org/10.1146/annurev-fluid-121108-145608). *Annual Review of Fluid Mechanics*, 42(1):413–438, 2010. [^morgado]: Rafael Morgado, Fernando A Oliveira, G George Batrouni, and Alex Hansen. [Relation between anomalous and normal diffusion in systems with memory](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.89.100601). *Phys Rev Lett*, 89(10):100601, 2002. [^kupferman]: Raz Kupferman. [Fractional kinetics in Kac–Zwanzig heat bath models](https://link.springer.com/article/10.1023/B:JOSS.0000003113.22621.f0). *Journal of statistical physics*, 114(1):291–326, 2004. [^kou]: SC Kou. [Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins](https://projecteuclid.org/journals/annals-of-applied-statistics/volume-2/issue-2/Stochastic-modeling-in-nanoscale-biophysics-Subdiffusion-within-proteins/10.1214/07-AOAS149.pdf). *The Annals of Applied Statistics*, pages 501–535, 2008. [^kneller]: Gerald R Kneller. [Generalized Kubo relations and conditions for anomalous diffusion: Physical insights from a mathematical theorem](https://doi.org/10.1063/1.3598483). *The Journal of Chemical physics*, 134(22), 2011. [^hohenegger2018]: Christel Hohenegger and **Scott A McKinley**. [Reconstructing complex fluid properties from the behavior of fluctuating immersed particles](https://doi.org/10.1137/17M1131660). *SIAM Journal on Applied Mathematics*, 78(4):2200–2226, 2018. [^hanke]: Martin Hanke. [The second fluctuation-dissipation theorem for the generalized Langevin equation](https://arxiv.org/abs/2507.17350). arxiv.org, 2025. [^goychuk2009]: Igor Goychuk. [Viscoelastic subdiffusion: From anomalous to normal.](https://doi.org/10.1103/PhysRevE.80.046125) *Physical Review E—Statistical, Nonlinear, and Soft Matter Physics*, 80(4):046125, 2009. [^goychuk2014]: Igor Goychuk, Vasyl O Kharchenko, and Ralf Metzler. [Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion](https://doi.org/10.1039/c4cp01234h). *Physical Chemistry Chemical Physics*, 16(31):16524–16535, 2014. [^bouzat]: Sebastián Bouzat. [Influence of molecular motors on the motion of particles in viscoelastic media](https://doi.org/10.1103/PhysRevE.89.062707). Physical Review E, 89(6):062707, 2014. [^gh-nonlinearity]: Nathen E Glatt-Holtz, David P Herzog, **Scott A McKinley**, and Hung D Nguyen. [The generalized Langevin equation with power-law memory in a nonlinear potential well](https://doi.org/10.1088/1361-6544/ab74af). *Nonlinearity.* 33:2280.