Mobility, response and transport in non-equilibrium coarse-grained models

Abstract: We investigate two different types of non-Markovian coarse-grained models extracted from a linear, non-equilibrium microscopic system, featuring a tagged particle coupled to underdamped oscillators. The first model is obtained by analytically ``integrating out'' the oscillators and the second is derived using projection operator techniques. We observe that these two models behave very differently when the tagged particle is exposed to external harmonic potentials or pulling forces. Most importantly, we find that the analytic model has a well defined friction kernel and can be used to extract work, consistent with the microscopic system, while the projection model corresponds to an effective equilibrium model, which cannot be used to extract work. We apply the analysis to two popular non-equilibrium systems, time-delay feedback control and the active Ornstein-Uhlenbeck process. Finally, we highlight that our study could have important consequences for dynamic coarse-graining of non-equilibrium systems going far beyond the linear systems investigated in this manuscript.

• The paper introduces two coarse-graining approaches (I-GLE and P-GLE) to analyze non-equilibrium dynamics by deriving explicit memory kernels and noise correlations.
• It shows that model responses in harmonic and sawtooth potentials reveal deviations from equilibrium predictions and effective temperature variations.
• Active microrheology experiments validate the I-GLE model by linking microscopic friction forces to transport phenomena in non-equilibrium conditions.

Mobility, Response, and Transport in Non-Equilibrium Coarse-Grained Models

This paper explores the behavior and properties of coarse-grained models in non-equilibrium systems, focusing on mobility, response, and transport phenomena. Two different coarse-grained approaches are analyzed: one derived analytically by integrating out oscillators and another using projection operator techniques. The study compares these approaches to better understand their implications for dynamic coarse-graining beyond linear systems, with detailed examinations of equilibrium and non-equilibrium conditions.

Microscopic Model and Dynamic Coarse-Graining

The microscopic system under investigation is a tagged particle interacting with several solvent particles. Three distinct variations—EQ, FEED, and OU—are studied, each presenting unique microscopic dynamics and interactions. The study's coarse-grained models are derived using the Generalized Langevin Equation (GLE) through two methods:

1. Integration Method (I-GLE): Here, solvent particles are analytically integrated out, providing explicit memory kernels and noise autocorrelation functions.
2. Projection Method (P-GLE): The Mori-Zwanzig formalism is applied, resulting in a model with a memory kernel fulfilling the fluctuation-dissipation theorem (2FDT).

   Figure 1: Memory kernel $K(t)$ and noise autocorrelation function $C_\eta(t)$ using the projection method ($p$) or the integration method ($I$) for the three different systems.

Both methods accurately capture some microscopic dynamics, as observed in identical velocity autocorrelation functions (VACFs) across models.

Figure 2: Velocity autocorrelation function $C_V(t)$ as extracted from different coarse-grained models compared to the theoretical prediction.

External Harmonic Potential

The study examines how coarse-grained models respond in harmonic potentials. In equilibrium (EQ), results align with the Boltzmann distribution. However, non-equilibrium systems (FEED, OU) show significant differences between the I-GLE and P-GLE models. The I-GLE captures complex behavior such as effective temperature dependence on potential strength and deviations from Boltzmann distribution, highlighting the intricate dynamics.

Figure 3: Position probability distribution $P(x)$ as extracted from different coarse-grained models for harmonic external potentials.

Figure 4: Effective temperature deviation $\Delta T = C_V(0) - C^{k=0}_V(0).</p></p> <h2 class='paper-heading' id='active-microrheology'>Active Microrheology</h2> <p>Active microrheological experiments reveal how colloids respond to constant external forces. The findings showcase distinct mobilities, confirming that I-GLE accurately connects to microscopic friction forces. Such insights underscore the model&#39;s practical relevance in simulating active particle behavior. <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2310-03565/FIG5.png" alt="Figure 5" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 5: Average velocity $\langle v \rangle$as response to a constant external pulling force$F_\text{ext}.$$</p></p> <h2 class='paper-heading' id='linear-response-and-first-fluctuation-dissipation-theorem'>Linear Response and First Fluctuation-Dissipation Theorem</h2> <p>The study further explores the unsteady response of systems using impulse forces. Equilibrium conditions adhere to the 1FDT, while non-equilibrium systems (e.g., FEED, OU) exhibit deviations. The I-GLE, reflecting true non-equilibrium properties, violates the 1FDT, indicating distinct dissipation rates and work potential. <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2310-03565/FIG6.png" alt="Figure 6" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 6: Linear time-dependent response$$\chi(t) = \langle v_0(t) \rangle$$to an impulse force.</p></p> <h2 class='paper-heading' id='sawtooth-potential-and-non-equilibrium-flow'>Sawtooth Potential and Non-Equilibrium Flow</h2> <p>Finally, the paper investigates work performance using an asymmetric sawtooth potential. While equilibrium systems show no movement, insights into non-equilibrium systems reveal differences. Specifically, the OU model showcases inherent flow, demonstrating its ability to perform work—an essential characteristic for biological and mechanical applications. <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2310-03565/FIG7.png" alt="Figure 7" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 7: Time-dependent position$$x(t)$ of an individual tracer in a sawtooth potential.

Conclusions: Implications for Coarse-Grained Models

The findings emphasize the critical need for precise dynamic coarse-graining methods to accurately capture non-equilibrium phenomena, reflected in transport properties, energy dissipation, and system thermodynamics. Looking ahead, expanding these methodologies to complex systems could greatly advance understanding and applications in diverse fields such as biology and socio-economic modeling. The study underscores the potential of enhancing reconstructing techniques to effectively bridge microscopic and coarse-grained models in non-equilibrium contexts.