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School of Mathematical Sciences

Thermodynamic signatures of collective behaviour in active matter

Supervisor: Dr Lennart Dabelow

Project description:

Context and significance

Active matter is a collective term for systems whose constituents can convert energy from their surroundings into directed motion. Prototypical examples are living organisms on all scales, from herds of mammals, schools of fish or colonies of ants down to the micro- and nanometre scale of archaea and bacteria. These systems can exhibit a variety of intriguing collective phenomena like clustering, swarming, and coordinated navigation, for example. Notably, such behaviour often emerges from purely local interactions between close-by agents, without sophisticated long-scale planning.

Within this project, we will explore the collective behaviour of active particles at the microscale. Examples include the aforementioned archaea and bacteria, but also artificial microswimmers like colloids with catalytic surfaces or nanorobots. A particular emphasis will be on understanding and characterizing such systems thermodynamically. At these scales, the particles are typically immersed in some fluid environment and are strongly affected by thermal fluctuations. This renders their motion erratic on long time scales, while it still exhibits the characteristic persistence of active matter on shorter time scales. Mathematically, their dynamics is thus described by systems of coupled, non-Markovian stochastic differential equations.

Furthermore, active-matter systems are inherently out of equilibrium as a result of the permanent conversion of energy. Hence the standard framework of macroscopic, equilibrium thermodynamics cannot be applied. Stochastic thermodynamics provides useful tools and concepts to characterize such systems instead, but the comprehensive and coherent description of active matter remains a highly active direction of current research. A major difficulty is that the microscopic processes underlying self-propulsion are usually not observable experimentally and hard to model consistently in theory.

We will build on recent insights [1] that link the observable irreversibility [2-3] to thermodynamic quantities, most notably the pressure. Irreversibility is defined as the log-ratio of forward- and backward-in-time path probabilities and thus quantifies how strongly microscopic particle trajectories break time-reversal symmetry. Connecting this purely dynamical measure to thermodynamic system properties can therefore be seen as an extension of the second law of thermodynamics to the nonequilibrium realm of active matter. Since the second law underlies essentially all applications in science and engineering that rely on thermodynamic processes, such an extension is not only of fundamental interest, but also of immediate practical relevance.


As a first step (year 1), we will investigate the aforementioned irreversibility measure and its relation to thermodynamics for more realistic models of active particles. Reference [1] focuses on spherical particles and Gaussian processes to model thermal fluctuations and self-propulsion forces. Both idealizations work phenomenologically in certain limiting cases, but miss important characteristics of, for example, many biological systems. Improved models may include, among others:

  • non-spherical particles [4],
  • non-Gaussian driving processes [5],
  • alignment interactions,
  • hydrodynamic interactions.

Modifying the setup along these lines changes the macroscopic phenomenology qualitatively, and this is expected to be reflected in the thermodynamic properties and their connection to irreversibility as well. Technically, these investigations will require a thorough analysis of the interaction kinematics and geometry as well as a different path-integral approach, with perturbation theory or large-deviations theory as potential starting points.

Later on (year 2–3), we will broaden the perspective by looking at different phases of these active-particle models in general. We will investigate the emergence of collective effects like formation of clusters, coordinated motion and phase coexistence, and will analyse how they manifest themselves in thermodynamic quantities such as entropy, pressure or heat capacity. This may include mixtures of different types of active and/or passive particles, in particular, as they are commonly found in biological environments.


Given existing collaborations with Nordita (Nordic Institute for Theoretical Physics) in Stockholm, the student will be encouraged to apply for and participate in their Visiting PhD Fellowship programme ( This will be an excellent opportunity to explore a different research environment and to get to know many other researchers, notably thanks to frequent multi-week scientific programmes hosted at Nordita.


Further information:

How to apply 
Entry requirements 
Fees and funding

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