Projects

Find here an overview of the proejcts of the 2nd funding period.

P1 - Time-local master equations for non-Markovian processes in quantum and classical open systems (Heinz-Peter Breuer, Johan Runeson and Michael Thoss)

The dynamics of open quantum systems is often governed by non-Markovian behavior and pronounced memory effects, featuring a backflow of information from the environment to the open system. Mathematically, this dynamics is commonly treated by means of master equations for the open system's density matrix, which can be derived employing suitable projection operator techniques. In the quantum regime there are basically two different variants of these techniques, namely the Nakajima-Zwanzig projection operator technique leading to a memory kernel master equation, and the time-convolutionless (TCL) projection operator technique yielding a time-local first order differential equation.

As part of the second funding period of the Research Unit "Reducing complexity of nonequilibrium systems" the present project is concerned with the general properties of TCL master equations describing classical and quantum non-Markovian processes in open systems. One of the central problems is the analysis of the mathematical structure of singularities of the time-dependent TCL generator. This is a challenging problem which has not yet been studied in general terms. We will develop a classification of the singularities and establish general conditions for their occurrence in physically relevant system-environment models. A further challenge is to investigate the impact of singularities on the performance of the TCL perturbation expansion in terms of the ordered cumulants of environmental correlations functions. Another problem of great relevance is the application to a recent formulation of nonequilibrium quantum thermodynamics, which employs time-local master equations to formulate work and heat exchange in nonequilibrium processes and, hence, requires a detailed discussion of the mathematical structure of the singularities and of their physical meaning and implication.

In addition to these mainly analytical works, we will also develop algorithms for the numerical determination of the time-dependent TCL generator. This is a problem of practical relevance as it allows to efficiently carry out long-time simulations. We will also compare the quality of the various approximations obtained from the time-local and from the memory-kernel master equation, and examine the performance of the TCL master equation in the classical regime of non-Markovianity. Finally, we will develop a mixed quantum-classical approach, combining the TCL master equation with a classical treatment of the slow degrees of freedom and a hopping process between energy surfaces of the effective Hamiltonian.

P2 - Charge transport and current-induced nonconservative forces in nanostructures (Michael Thoss, Heinz-Peter Breuer & Gerhard Stock)

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P3 - Project will not be continued

P4 - Coarse-graining from the Liouville equation to fluid dynamics (Tanja Schilling, Lars Pastewka & Kerstin Falk)

The theory of hydrodynamics is based on conservation laws and constitutive relations. While the conservation laws are universal, the constitutive relations are specific to system and external conditions. In the proposed project we will derive the evolution equations for the mass, energy and momentum density field from the underlying Hamiltonian dynamics.

We will complement the theoretical work by large-scale computer simulations of liquids close to the glass transition and polymers under external driving in order to quantify the effects of memory and non-equilibrium response. Our goal is to propose extensions to the theory of hydrodynamics that can be applied to systems with interfering relaxation time-scales and to systems under time-dependent driving.

P5 - Data-driven Markov modeling of nonequilibrium processes (Gerhard Stock & Michael Thoss)

Te project is concerned with the extension of the theory and the numerical implementation of Markov state models (MSM) to treat non-Markovian and nonequilibrium data. To model non-Markovian data by a generalized master equation, we will employ the quasi-MSM ansatz and also consider time-convolutionless formulations. We will study various approximations to evaluate the associated memory kernel matrix and time-dependent rate matrix, respectively, considering in particular
the case of noisy input data. To model the nonequilibrium response of photoswitchable proteins, we will construct a network of weakly interacting MSMs representing contact clusters, which mediate a multi-step structural reorganization process consisting of cooperative conformational transitions within a cluster and of the communication between clusters. Moreover, we will develop an advanced dynamical clustering scheme to construct metastable states, which lumps
microstates according to their transition probability distribution, invokes geometrical information, and is suited for nonequilibrium data.

Markov-Type State Models to Describe Non-Markovian Dynamics
S. Sartore, F. Teichmann, G. Stock, J. Chem. Theory Comput. 21, 2757 (2025)

P6 - P6: Dzubiella, Schilling, Moseler Feedback-controlled relaxation pathways of responsive colloids (Joachim Dzubiella, Tanja Schilling & Michael Moseler )

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P7 - Nonequilibrium functional dynamics of proteins (Gerhard Stock, Steffen Wolf & Aljaz Godec)

The project is concerned with the binding and unbinding of ligands to proteins and the resulting ligand-induced conformation transitions in intramolecular signaling. To explore various nonequilibrium conditions –such as the perturbation by light, mechanical stress and ligand binding– nonequilibrium MD simulations of photoinduced conformational change as well as dissipation-corrected targeted MD (dcTMD) will be employed. dcTMD will be benchmarked against a newly available theory-internal protein-ligand reference system, to evaluate its capabilities and shortcomings in finding ligand diffusion pathways and intermediate binding sites.

Moreover, we will use dcTMD to understand deviations in bath dynamics from the equilibrium imposed by external driving and how they manifest in changes in dissipation as well as in fluctuations.

To further explore the role of contact clusters in protein allosteric communication, we will extend our previous analysis of the PDZ3 domain to a broader class of proteins, including the tetracycline repressor TetR and heat shock protein 90. For systems with allosteric transition times on the micro- to millisecond scale, we will construct low-dimensional biasing coordinates based on relevant contact clusters and apply enhanced sampling techniques such as dcTMD and
metadynamics. On a more coarse-grained theoretical level, we will conduct full mechanical-response analyses using a nonlinear, reversibly dissociable elastic network model. This approach will enable us to examine key aspects of allosteric communication, including nonlinearity, directionality, and specificity, which is also studied in project P5.

Nonequilibrium friction and free energy estimates for kinetic coarse-graining—Driven particles in responsive media
S. Milster, J. Dzubiella, G. Stock, S. Wolf
J. Chem. Phys. 162, 154113 (2025)

Contact Cluster Modeling of Allosteric Communication in PDZ Domains
E. Dorbath, F. Rudolf, A. Gulzar, G. Stock
J. Phys. Chem. B 130, 1121 (2026)

P8 - Towards a continuum theory of sheared fluids under stong confinement (Kerstin Falk, Steffen Wolf & Michael Moseler)

Project P8 "Boundary lubrication: Toward a continuum theory of sheared fluids under strong confinement" is aimed at finding a comprehensive continuum description of complex tribological contacts in the boundary lubrication regime. The latter is characterized by extreme non-equilibrium states of the lubricant due to very high pressures, shear rates and strong confinement and a non-linear response of the lubricant to the external driving. The challenges for a continuum theory of boundary lubrication are in formulating physics-based constitutive laws for the non-linear lubricant transport properties, and in applying a continuum framework to conditions under which the traditional hydrodynamics theory might fail. In this project, these challenges are tackled in a combination of large-scale molecular dynamics simulations, continuum numerical calculations based on the Reynolds lubrication equations and mechanistic theoretical modelling. During the first funding period of the research unit, we showed that the traditional hydrodynamic description of lubricated contacts can be extended to channels with nanometer-thin constrictions if the non-linear dependence of viscosity and wall slip on the local shear stress are taken properly into account [1]. This was done by extracting constitutive laws for viscosity and wall slip in the non-linear regime from a large parametric molecular dynamics simulations study. These viscosity and slip laws, which do not depend on the lubrication gap size, were integrated into a continuum code based on the Reynolds lubrication equations and successfully used for modeling of systems with minimum gap sizes down to about 2nm and Gigapascal pressures. For thinner lubricant films, we observed important deviations, which are now considered in a systematic way during the second project period. Parameter studies of lubricant transport in sheared contacts with parallel walls and with geometries including roughness will be performed, and focus on the transitions between nanometer gap heights, monolayer lubrication and dry contact will be laid. Gap height-dependent constitutive equations for relevant lubricant properties, such as viscosity, diffusion, fluid/solid wall slip and disjoining pressure will be extracted from the parallel wall simulations. These constitutive equations will be implemented in an existing continuum Reynolds code, which will be extended to include the monolayer lubrication regime. The code will then be tested against the molecular dynamics data on the large-scale tribological contact including roughness.

[1] Codrignani, A., Peeters, S., Holey, H., Stief, F., Savio, D., Pastewka, L., Moras, G., Falk, K. and Moseler, M.: Toward a continuum description of lubrication in highly pressurized nanometer-wide constrictions: The importance of accurate slip laws. Science Advances, 9(48), 2023, 10.1126/sciadv.adi2649

P9 - Project will not be continued

P10 - Thermophoresis from first principles (Aljaz Godec, Tanja Schilling & Joachim Dzubiella)

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P11 - A mixed quantum-classical approach to vibrational energy transport (Johan E. Runeson & Gerhard Stock)

Vibrational energy transport in proteins is important for a variety of nonequilibrium biological processes, but very difficult to describe accurately. Standard classical molecular dynamics simulations neglect quantum effects, such as zero-point energy, leading to incorrect rates and pathways in comparison to a quantum description. However, fully quantum calculations are not feasible for large atomistic systems. The aim of this project is to find the appropriate intermediate level of theory — a mixed quantum-classical theory — to accurately and efficiently describe vibrational energy flows in biomolecules.

The project will involve modelling and interpreting recent experiments that have measured the flow of energy in peptides on the level of individual amino acids. These experiments inject vibrational energy into the molecule by photoexciting a donor group (red in the figure below). The excess vibrational energy can then flow along the backbone as well as across contacts. In order to understand the relative importance of different pathways in the measured signals, we need to accurately calculate the rates for each type of transport mode. The project will unravel the origin and magnitude of quantum corrections to these rates using a variety of methods, including fully quantum calculations, perturbative approaches, and explicit mixed quantum-classical methods such as surface hopping.

Figure source: Buchenberg, Leitner and Stock, J. Phys. Chem. Lett. 7, 25 (2016)