Periodic Reporting for period 1 - IDAM (Inverse Design of Active Matter)
Reporting period: 2023-09-01 to 2025-08-31
Active matter (AM) represents a fascinating system where continuous energy flow enables persistent dynamics like locomotion in inert materials and may lead to fascinating collective behaviors resembling those of living agents such as bird flocks or bacterial vortices. In particular, collective order in AM stands to enable the realization of a new generation of materials with customizable dynamic functions such as targeted transport, self-healing, sensing among others, and is a rapidly expanding field both at the theoretical and experimental level. However, how these collective behaviors emerge and how they correlate to the underlying energy flow remains an open challenge, especially at the small scales where top-down manipulation is often impracticable or unfeasible. It further bears important implication to living systems which stand as a paradigmatic example of how finely controlled programs of energy flow result in robust dynamical functions and diverse collective behavior.
In IDAM we aim to address key areas of this question via the study of a more general class of active matter systems at the particle and field level via discovered and directed strategies for collective control. This approach follows in the spirit of inverse design strategies where desirable collective structures are realized as the stable state from an otherwise disorganized system. These efforts typically combine theory and numerics to formulate an optimization problem from sound physical principles and solve it via a combination of numerics and simulations. For instance, colloidal structures with specific structures (e.g. crystals), materials properties (elastic response, wave propagation etc) or condensed behavior (fluid porosity) have been successfully designed this way via equilibrium, free-energy principles.
However, extending such an approach to AM is non-trivial given its inherent non-equilibrium character via continuous energy consumption. In this project we therefore advance large deviation theory (LDT) as a powerful framework to discover collective phase transitions in AM via control of the underlying dynamics. In particular, LDT allows to alter a system’s dynamics via calculation of internal dynamic observables such as a time average by biasing (hence directing) energy-flows. Indeed, similar approaches to systems of self-propelling particles show that biasing with respect to its energy flow resulted in the formation of well known collective phases such as clustering and flocking of particles, which respectively, represent low and high forms of energy flow in these systems. In IDAM our main goal is to develop a general LDT framework which we apply to a system of particles with continuous size pulsation to explore their emergent collective phases.
We additionally look at a more general class of control problems and consider the case of phase transformations between an initial and final state in finite time. Examples of such processes take place, for instance, in the folding-unfolding transition of a DNA-hair pin motif upon stretching or release from an externally controlled force, or the optimal flipping of a spin state in a magnetic material. From the second law of thermodynamics, we know such transformation energy cost is minimal in the quasistatic limit i.e. as the protocol duration goes to infinity and given by a free-energy difference. However, realistic experimental transitions must necessarily take place in finite time and therefore imply an inevitable energy loss when carried out. Nonetheless, one may consider a protocol which minimizes such a loss and which, from this perspective, represents an optimal control between the two state transition process.
IDAM therefore develops and explores two distinct strategies of control via the following major objectives:
1) Development of a numerical LDT framework which to study collective phase transitions of dynamical systems like active matter models
2) Application of the framework to a system of pulsating active particles in context of collective phase transitions and consideration to continuum active matter models
3) Development of an optimal control optimizer in finite time from a variational formulation of a minimum dissipation protocols
4) Application of a liquid-state transformation via tuning of particle size and interactions in a paradigmatic model of liquids with Lennard-Jones interactions
In IDAM we aim to address key areas of this question via the study of a more general class of active matter systems at the particle and field level via discovered and directed strategies for collective control. This approach follows in the spirit of inverse design strategies where desirable collective structures are realized as the stable state from an otherwise disorganized system. These efforts typically combine theory and numerics to formulate an optimization problem from sound physical principles and solve it via a combination of numerics and simulations. For instance, colloidal structures with specific structures (e.g. crystals), materials properties (elastic response, wave propagation etc) or condensed behavior (fluid porosity) have been successfully designed this way via equilibrium, free-energy principles.
However, extending such an approach to AM is non-trivial given its inherent non-equilibrium character via continuous energy consumption. In this project we therefore advance large deviation theory (LDT) as a powerful framework to discover collective phase transitions in AM via control of the underlying dynamics. In particular, LDT allows to alter a system’s dynamics via calculation of internal dynamic observables such as a time average by biasing (hence directing) energy-flows. Indeed, similar approaches to systems of self-propelling particles show that biasing with respect to its energy flow resulted in the formation of well known collective phases such as clustering and flocking of particles, which respectively, represent low and high forms of energy flow in these systems. In IDAM our main goal is to develop a general LDT framework which we apply to a system of particles with continuous size pulsation to explore their emergent collective phases.
We additionally look at a more general class of control problems and consider the case of phase transformations between an initial and final state in finite time. Examples of such processes take place, for instance, in the folding-unfolding transition of a DNA-hair pin motif upon stretching or release from an externally controlled force, or the optimal flipping of a spin state in a magnetic material. From the second law of thermodynamics, we know such transformation energy cost is minimal in the quasistatic limit i.e. as the protocol duration goes to infinity and given by a free-energy difference. However, realistic experimental transitions must necessarily take place in finite time and therefore imply an inevitable energy loss when carried out. Nonetheless, one may consider a protocol which minimizes such a loss and which, from this perspective, represents an optimal control between the two state transition process.
IDAM therefore develops and explores two distinct strategies of control via the following major objectives:
1) Development of a numerical LDT framework which to study collective phase transitions of dynamical systems like active matter models
2) Application of the framework to a system of pulsating active particles in context of collective phase transitions and consideration to continuum active matter models
3) Development of an optimal control optimizer in finite time from a variational formulation of a minimum dissipation protocols
4) Application of a liquid-state transformation via tuning of particle size and interactions in a paradigmatic model of liquids with Lennard-Jones interactions
For the first part, a numerical framework to compute large deviation functions was developed that is general for a wide class of active matter models. The core of the code is based on a population dynamics algorithm which enables the computation of a scaled cumulative generating function (SCGF)—a key function in LDT that, in essence, helps quantify the distance of a desired dynamic behavior away from its typical (average) behavior in time. This property is encoded via i) a dynamic observable that is judiciously chosen to reflect a dynamic behavior of interest and ii) a bias parameter that effectively tilts the dynamics such that this observable displays values above its typical value. Additionally, the code offers a controlled-dynamic feature that helps improve SCGF convergence, hence accuracy of the sampling and which we applied via an `on-the-fly’ iterative procedure.
Using this numerical approach, we studied the collective dynamics of a pulsating active matter (PAM) system in dense conditions. In particular, we wanted to bias the energy flow in PAM such that synchronization could be induced and therefore collective effects. This consideration resulted in an equivalent formulation of two dynamic observables: 1) a global order parameter, which, essentially informs on the uniformity of particle size and 2) a local order parameter which only focuses on the uniformity of nearest neighbors. Our main results show that biasing these systems with respect to the global order parameter resulted in the formation of globally cycling and arrested states. This latter was particularly remarkable because it demonstrated that dynamic transformations can be controlled by how packing properties of the particles, via geometric frustration, and which follow from the natural dynamics of the system (without bias). On the other hand, biasing with respect to the local order parameter conserved the original phase results but also uncovered a new phase in the form of propagation waves at intermediate biases. These finding therefore uncovered the physical role of bias as encoding for strength and range of synchronization in the system. These results were published in the flagship Physical Review Letters journal, further receiving an Editor’s Choice upon publication.
The second major work involved the development and application of optimal control problems in a paradigmatic model of a Lennard-Jones fluid. In particular, we considered the case where we minimize the total dissipation for a transition between a low density and high density phase via changes in the particles’ size and interaction strength. To this end we adapted a linear response framework that effectively maps such a protocol transformation as a displacement within a `thermodynamic metric’ where the optimal paths manifest as geodesics of this curved space. The key ingredients of the approach were 1) the computation of the thermodynamic space which is obtained from simulation of time correlation functions with respect to parameter changes and 2) finding the optimal protocol path in this space via a numerical optimization method. Both these tasks were achieved via development of an in-house code via molecular dynamics simulation and the application of an implicit, “string” optimizer method.
Results included the elucidation of the metric space via the friction tensor, which similar to work on a many-body Ising model, showed it diverges as it approaches phase separated regions. In particular, the friction tensor revealed that, generally, friction increases with increasing density and is mostly uniform up to the regions of phase separation. Optimal control in this space revealed that optimal protocols follow curved paths away from regions where the thermodynamic metric quickly diverges corresponding to phase separation in the system. Physically it implies that a liquid transition in finite time achieves minimum dissipation by conserving a homogenous profile and avoiding phase separation by staying above the critical region. Comparing energy performance between a naïve protocol—i.e. a linear change of parameters connecting initial and final state—and the optimal path showed that indeed the optimal protocol significantly minimized energy loss becoming more significant in regions near phase separation. Remarkably, optimal path predictions remained valid even for short protocol durations technically outside the linear assumptions of the method, and highlights the overall robustness of the approach. The manuscript for this work is currently under preparation and will be submitted for review in the Journal of Chemical Physics.
Using this numerical approach, we studied the collective dynamics of a pulsating active matter (PAM) system in dense conditions. In particular, we wanted to bias the energy flow in PAM such that synchronization could be induced and therefore collective effects. This consideration resulted in an equivalent formulation of two dynamic observables: 1) a global order parameter, which, essentially informs on the uniformity of particle size and 2) a local order parameter which only focuses on the uniformity of nearest neighbors. Our main results show that biasing these systems with respect to the global order parameter resulted in the formation of globally cycling and arrested states. This latter was particularly remarkable because it demonstrated that dynamic transformations can be controlled by how packing properties of the particles, via geometric frustration, and which follow from the natural dynamics of the system (without bias). On the other hand, biasing with respect to the local order parameter conserved the original phase results but also uncovered a new phase in the form of propagation waves at intermediate biases. These finding therefore uncovered the physical role of bias as encoding for strength and range of synchronization in the system. These results were published in the flagship Physical Review Letters journal, further receiving an Editor’s Choice upon publication.
The second major work involved the development and application of optimal control problems in a paradigmatic model of a Lennard-Jones fluid. In particular, we considered the case where we minimize the total dissipation for a transition between a low density and high density phase via changes in the particles’ size and interaction strength. To this end we adapted a linear response framework that effectively maps such a protocol transformation as a displacement within a `thermodynamic metric’ where the optimal paths manifest as geodesics of this curved space. The key ingredients of the approach were 1) the computation of the thermodynamic space which is obtained from simulation of time correlation functions with respect to parameter changes and 2) finding the optimal protocol path in this space via a numerical optimization method. Both these tasks were achieved via development of an in-house code via molecular dynamics simulation and the application of an implicit, “string” optimizer method.
Results included the elucidation of the metric space via the friction tensor, which similar to work on a many-body Ising model, showed it diverges as it approaches phase separated regions. In particular, the friction tensor revealed that, generally, friction increases with increasing density and is mostly uniform up to the regions of phase separation. Optimal control in this space revealed that optimal protocols follow curved paths away from regions where the thermodynamic metric quickly diverges corresponding to phase separation in the system. Physically it implies that a liquid transition in finite time achieves minimum dissipation by conserving a homogenous profile and avoiding phase separation by staying above the critical region. Comparing energy performance between a naïve protocol—i.e. a linear change of parameters connecting initial and final state—and the optimal path showed that indeed the optimal protocol significantly minimized energy loss becoming more significant in regions near phase separation. Remarkably, optimal path predictions remained valid even for short protocol durations technically outside the linear assumptions of the method, and highlights the overall robustness of the approach. The manuscript for this work is currently under preparation and will be submitted for review in the Journal of Chemical Physics.
The project helped showcase two distinct aspects of control in soft and active matter systems. For the first, we demonstrated that an approach with LDT allows to direct and discover collective dynamic transitions via energy flow considerations. It furthermore highlighted the role that geometric frustration plays in dense active matter, which in our case served as a simple case of an experimentally accessible control parameter—changing the box geometry—in driving distinct collective transitions. It also adds to the list of many-body problems that are now accessible via LDT and sets the stage for exploring dynamic phase transitions in many other complex interacting systems to workers in the AM and soft-matter communities more broadly.
The second approach introduces ideas of optimal control in a paradigmatic model of liquids like the Lennard-Jones system. We showed that even in condensed interacting systems optimal control can inform and guide state transformation via non-trivial, optimal protocols that minimize energy consumption. Importantly it shows that avoiding phase separation is a key physical requirement to prevent unnecessary energy dissipation and that it requires attenuation of both size and interaction strength parameters in the Lennard-Jones fluid. This work demonstrates the value of optimal protocols in condensed matter systems and showcases its use in minimizing energy consumption in driving state transformations in interacting, many-body system.
Together, it is expected both works will further motivate and aid the design and synthesis of novel condensed and active matter materials, both from a phase transition point of view---via control of structural properties---and a dynamical behavior perspective with LDT. Such strategies hold great potential in optimizing fundamental thermodynamic losses or uncovering simple and accessible control parameters such as e.g. changing a box geometry to switch between overall collective behaviors. These findings are of particular interest to transformations that involve energy processes including materials for energy storage, transfer or the enabling of responsive behavior for life-mimetic systems.
The second approach introduces ideas of optimal control in a paradigmatic model of liquids like the Lennard-Jones system. We showed that even in condensed interacting systems optimal control can inform and guide state transformation via non-trivial, optimal protocols that minimize energy consumption. Importantly it shows that avoiding phase separation is a key physical requirement to prevent unnecessary energy dissipation and that it requires attenuation of both size and interaction strength parameters in the Lennard-Jones fluid. This work demonstrates the value of optimal protocols in condensed matter systems and showcases its use in minimizing energy consumption in driving state transformations in interacting, many-body system.
Together, it is expected both works will further motivate and aid the design and synthesis of novel condensed and active matter materials, both from a phase transition point of view---via control of structural properties---and a dynamical behavior perspective with LDT. Such strategies hold great potential in optimizing fundamental thermodynamic losses or uncovering simple and accessible control parameters such as e.g. changing a box geometry to switch between overall collective behaviors. These findings are of particular interest to transformations that involve energy processes including materials for energy storage, transfer or the enabling of responsive behavior for life-mimetic systems.