Welcome to my webpage!

The research in my group is focused on understanding complex systems away from thermal equilibrium and developing fundamental theories to describe them. I’m currently mostly interested in

  • Granular and soft matter: jamming, packings of non-spherical particles, shear flows.
  • Transport processes: noise-induced activation, anomalous diffusion, turbulence.

We apply a wide-range of mathematical methods from statistical mechanics (ensemble approaches, large deviation theory) and stochastic processes (SDEs, path-integrals, Feynman-Kac theorem). For more information on my research, please take a look at the list of publications. My contact details can be found under the team page.

Prospective PhD students and postdocs

Enquiries for possible PhD and postdoctoral projects are always welcome! UK funding for research projects is very limited unfortunately. Ideally, you should be able to obtain funding from your home country, which is possible, e.g., for Chinese students through the CSC scheme.

Fully funded Queen Mary PhD studentships are advertised around November/December, see the information on this link.


  • EPSRC First Grant: Optimizing particle packings by shape variation (£125,000). From 09/2014 to 08/2016.

Selected recent publications

  1. Weak Galilean invariance as a selection principle for coarse-grained diffusive models, A. Cairoli, R. Klages, and A. Baule
    Proc. Nat. Acad. Sci. USA (in print)
  2. Edwards statistical mechanics for jammed granular matter, A. Baule, F. Morone, H. Herrmann, and H. A. Makse
    Reviews of Modern Physics 90, 015006 (2018)
  3. Shape universality classes in the random sequential adsorption of non-spherical particles, A. Baule
    Physical Review Letters 119, 028003 (2017)
  4. Anomalous Processes with General Waiting Times: Functionals and Multi-point structure, A. Cairoli and A. Baule
    Physical Review Letters 115, 110601 (2015)
  5. Mean-field theory of random close packings of axisymmetric particles, A. Baule, R. Mari, L. Bo, L. Portal, and H. A. Makse
    Nature Communications 4, 2194 (2013)


April 2018

  • Weak Galilean invariance as a selection principle for coarse-grained diffusive models, A. Cairoli, R. Klages, and A. Baule
    Proc. Nat. Acad. Sci. USA (in print)

  • Abstract: How does the mathematical description of a system change in different reference frames? Galilei first addressed this fundamental question by formulating the famous principle of Galilean invariance. It prescribes that the equations of motion of closed systems remain the same in different inertial frames related by Galilean transformations, thus imposing strong constraints on the dynamical rules. However, real world systems are often described by coarse-grained models integrating complex internal and external interactions indistinguishably as friction and stochastic forces. Since Galilean invariance is then violated, there is seemingly no alternative principle to assess a priori the physical consistency of a given stochastic model in different inertial frames. Here, starting from the Kac-Zwanzig Hamiltonian model generating Brownian motion, we show how Galilean invariance is broken during the coarse graining procedure when deriving stochastic equations. Our analysis leads to a set of rules characterizing systems in different inertial frames that have to be satisfied by general stochastic models, which we call "weak Galilean invariance". Several well-known stochastic processes are invariant in these terms, except the continuous-time random walk for which we derive the correct invariant description. Our results are particularly relevant for the modelling of biological systems, as they provide a theoretical principle to select physically consistent stochastic models prior to a validation against experimental data.

July 2017

  • Edwards statistical mechanics for jammed granular matter, A. Baule, F. Morone, H. Herrmann, and H. A. Makse
    Reviews of Modern Physics 90, 015006 (2018)

  • Abstract:In 1989, Sir Sam Edwards made the visionary proposition to treat jammed granular materials using a volume ensemble of equiprobable jammed states in analogy to thermal equilibrium statistical mechanics, despite their inherent athermal features. Since then, the statistical mechanics approach for jammed matter -- one of the very few generalizations of Gibbs-Boltzmann statistical mechanics to out of equilibrium matter -- has garnered an extraordinary amount of attention by both theorists and experimentalists. Its importance stems from the fact that jammed states of matter are ubiquitous in nature appearing in a broad range of granular and soft materials such as colloids, emulsions, glasses, and biomatter. Indeed, despite being one of the simplest states of matter -- primarily governed by the steric interactions between the constitutive particles -- a theoretical understanding based on first principles has proved exceedingly challenging. Here, we review a systematic approach to jammed matter based on the Edwards statistical mechanical ensemble. We discuss the construction of microcanonical and canonical ensembles based on the volume function, which replaces the Hamiltonian in jammed systems. The importance of approximation schemes at various levels is emphasized leading to quantitative predictions for ensemble averaged quantities such as packing fractions and contact force distributions. An overview of the phenomenology of jammed states and experiments, simulations, and theoretical models scrutinizing the strong assumptions underlying Edwards' approach is given including recent results suggesting the validity of Edwards ergodic hypothesis for jammed states. A theoretical framework for packings whose constitutive particles range from spherical to non-spherical shapes like dimers, polymers, ellipsoids, spherocylinders or tetrahedra, hard and soft, frictional, frictionless and adhesive, monodisperse and polydisperse particles in any dimensions is discussed providing insight into an unifying phase diagram for all jammed matter. Furthermore, the connection between the Edwards' ensemble of metastable jammed states and metastability in spin-glasses is established. This highlights that the packing problem can be understood as a constraint satisfaction problem for excluded volume and force and torque balance leading to a unifying framework between the Edwards ensemble of equiprobable jammed states and out-of-equilibrium spin-glasses.

July 2017

  • Shape universality classes in the random sequential adsorption of non-spherical particles, A. Baule
    Physical Review Letters 119, 028003 (2017)

  • Abstract: Random sequential adsorption (RSA) of particles of a particular shape is used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the observed algebraic time dependence of the asymptotic jamming coverage $\sim t^{-\nu}$ as $t\to\infty$. However, the exact value of the exponent $\nu$ is not known apart from the simplest case of the RSA of monodisperse spheres adsorbed on a line (Renyi's seminal `car parking problem'), where $\nu=1$ can be derived analytically. Empirical simulation studies have conjectured on a case-by-case basis that for general non-spherical particles $\nu=1/(d+\tilde{d})$, where $d$ denotes the dimension of the domain and $\tilde{d}$ the number of orientational degrees of freedom of a particle. Here, we solve this long standing problem analytically for the $d=1$ case --- the `Paris car parking problem'. We prove in particular that the scaling exponent depends on particle shape, contrary to the original conjecture, and, remarkably, falls into two universality classes: (i) $\nu=1/(1+\tilde{d}/2)$ for shapes with a smooth contact distance, e.g., ellipsoids; (ii) $\nu=1/(1+\tilde{d})$ for shapes with a singular contact distance, e.g., spherocylinders and polyhedra. The exact solution explains in particular why many empirically observed scalings fall in between these two limits.

Jan 2017

GeYOPP smartphone app goes live for iOS and Android. Draw arbitrary 2D shapes and perform a packing experiment on your smartphone to find the shape that packs the densest! More information and download.

July 2016

Together with Peter Sollich (King’s College) and Tomaso Aste (UCL) I organized the International Workshop on Jamming and Granular Matter, a satellite meeting of StatPhys 26. The list of invited speakers and more details can be found on the workshop homepage.

September 2015

  • Anomalous Processes with General Waiting Times: Functionals and Multi-point structure, A. Cairoli and A. Baule
    Physical Review Letters 115, 110601 (2015).

    Abstract: Many transport processes in nature exhibit anomalous diffusive properties with non-trivial scaling of the mean square displacement, e.g., diffusion of cells or of biomolecules inside the cell nucleus, where typically a crossover between different scaling regimes appears over time. Here, we investigate a class of anomalous diffusion processes that is able to capture such complex dynamics by virtue of a general waiting time distribution. We obtain a complete characterization of such generalized anomalous processes, including their functionals and multi-point structure, using a representation in terms of a normal diffusive process plus a stochastic time change. In particular, we derive analytical closed form expressions for the two-point correlation functions, which can be readily compared with experimental data.

July 2015

  • Adhesive loose packings of small dry particles, W. Liu, S. Li, A. Baule, and H. A. Makse
    Soft Matter (2015).

    Abstract: We explore adhesive loose packings of small dry spherical particles of micrometer size using 3D discrete-element simulations with adhesive contact mechanics and statistical ensemble theory. A dimensionless adhesion parameter (Ad) successfully combines the effects of particle velocities, sizes and the work of adhesion, identifying a universal regime of adhesive packings for Ad > 1. The structural properties of the packings in this regime are well described by an ensemble approach based on a coarse-grained volume function that includes the correlation between bulk and contact spheres. Our theoretical and numerical results predict: (i) an equation of state for adhesive loose packings that appear as a continuation from the frictionless random close packing (RCP) point in the jamming phase diagram and (ii) the existence of an asymptotic adhesive loose packing point at a coordination number Z = 2 and a packing fraction ϕ = 1/2^3. Our results highlight that adhesion leads to a universal packing regime at packing fractions much smaller than the random loose packing (RLP), which can be described within a statistical mechanical framework. We present a general phase diagram of jammed matter comprising frictionless, frictional, adhesive as well as non-spherical particles, providing a classification of packings in terms of their continuation from the spherical frictionless RCP.

  • Langevin formulation of a subdiffusive continuous-time random walk in physical time, A. Cairoli and A. Baule
    Physical Review E 92, 012102 (2015).

    Abstract: Systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. One of the most common models to describe such subdiffusive dynamics is the continuous-time random walk (CTRW). Stochastic trajectories of a CTRW can be described in terms of the subordination of a normal diffusive process by an inverse Lévy-stable process. Here, we propose an equivalent Langevin formulation of a force-free CTRW without subordination. By introducing a different type of non-Gaussian noise, we are able to express the CTRW dynamics in terms of a single Langevin equation in physical time with additive noise. We derive the full multipoint statistics of this noise and compare it with the scaled Brownian motion (SBM), an alternative stochastic model describing subdiffusive dynamics. Interestingly, these two noises are identical up to the second order correlation functions, but different in the higher order statistics. We extend our formalism to general waiting time distributions and force fields and compare our results with those of the SBM. In the presence of external forces, our proposed noise generates a different class of stochastic processes, resembling a CTRW but with forces acting at all times.

June 2015

I joined the editorial board of Heliyon, Elsevier’s new multi-disciplinary open access journal. Heliyon publishes “technically sound content across all disciplines”.