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We develop a theoretical description of the topological disentanglement occurring when torus knots reach the ends of a semi-flexible polymer under tension. These include decays into simpler knots and total unknotting. The minimal number of crossings and the minimal knot contour length are the topological invariants playing a key role in the model. The crossings behave as particles diffusing along the chain and the application of appropriate boundary conditions at the ends of the chain accounts for the knot disentanglement. Starting from the number of particles and their positions, suitable rules allow reconstructing the type and location of the knot moving on the chain. Our theory is extensively benchmarked with corresponding Molecular Dynamics simulations and the results show a remarkable agreement between the simulations and the theoretical predictions of the model.
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In this review we provide an organized summary of the theoretical and computational results which are available for polymers subject to spatial or topological constraints. Because of the interdisciplinary character of the topic, we provide an accessible, non-specialist introduction to the main topological concepts, polymer models, and theoretical/computational methods used to investigate dense and entangled polymer systems. The main body of our review deals with: (i) the effect that spatial confinement has on the equilibrium topological entanglement of one or more polymer chains and (ii) the metric and entropic properties of polymer chains with fixed topological state. These problems have important technological applications and implications for the life-sciences. Both aspects, especially the latter, are amply covered. A number of selected open problems are finally highlighted.
In this work a new strategy is proposed in order to build analytic and microscopic models of fluctuating polymer rings subjected to topological constraints. The topological invariants used to fix these constraints belong to a wide class of the so-called numerical topological invariants. For each invariant it is possible to derive a field theory that describes the statistical behavior of knotted and linked polymer rings following a straightforward algorithm. The treatment is not limited to the partition function of the system, but it allows also to express the expectation values of general observables as field theory amplitudes. Our strategy is illustrated taking as examples the Gauss linking number and a topological invariant belonging to a class of invariants due to Massey. The consistency of the new method developed here is checked by reproducing a previous field theoretical model of two linked polymer rings. After the passage to field theory, the original topological constraints imposed on the fluctuating paths of the polymers become constraints over the configurations of the topological fields that mediate the interactions of topological origin between the monomers. These constraints involve quantities like the cross-helicity which are of interest in other disciplines, like for instance in modeling the solar magnetic field. While the calculation of the vacuum expectation values of generic observables remains still challenging due to the complexity of the problem of topological entanglement in polymer systems, we succeed here to reduce the evaluation of the moments of the Gauss linking number for two linked polymer rings to the computation of the amplitudes of a free field theory.
The coupling of active, self-motile particles to topological constraints can give rise to novel non-equilibrium dynamical patterns that lack any passive counterpart. Here we study the behavior of self-propelled rods confined to a compact spherical manifold by means of Brownian dynamics simulations. We establish the state diagram and find that short active rods at sufficiently high density exhibit a glass transition toward a disordered state characterized by persistent self-spinning motion. By periodically melting and revitrifying the spherical spinning glass, we observe clear signatures of time-dependent aging and rejuvenation physics. We quantify the crucial role of activity in these non-equilibrium processes, and rationalize the aging dynamics in terms of an absorbing-state transition toward a more stable active glassy state. Our results demonstrate both how concepts of passive glass phenomenology can carry over into the realm of active matter, and how topology can enrich the collective spatiotemporal dynamics in inherently non-equilibrium systems.
Anomalous diffusion, manifest as a nonlinear temporal evolution of the position mean square displacement, and/or non-Gaussian features of the position statistics, is prevalent in biological transport processes. Likewise, collective behavior is often observed to emerge spontaneously from the mutual interactions between constituent motile units in biological systems. Examples where these phenomena can be observed simultaneously have been identified in recent experiments on bird flocks, fish schools and bacterial swarms. These results pose an intriguing question, which cannot be resolved by existing theories of active matter: How is the collective motion of these systems affected by the anomalous diffusion of the constituent units? Here, we answer this question for a microscopic model of active Levy matter -- a collection of active particles that perform superdiffusion akin to a Levy flight and interact by promoting polar alignment of their orientations. We present in details the derivation of the hydrodynamic equations of motion of the model, obtain from these equations the criteria for a disordered or ordered state, and apply linear stability analysis on these states at the onset of collective motion. Our analysis reveals that the disorder-order phase transition in active Levy matter is critical, in contrast to ordinary active fluids where the phase transition is, instead, first-order. Correspondingly, we estimate the critical exponents of the transition by finite size scaling analysis and use these numerical estimates to relate our findings to known universality classes. These results highlight the novel physics exhibited by active matter integrating both anomalous diffusive single-particle motility and inter-particle interactions.
Long linear polymers in strongly disordered media are well described by self-avoiding walks (SAWs) on percolation clusters. The length-distribution of these SAWs encompasses to distinct averages, viz. the averages over cluster- and SAW-conformations. For the latter average, there are two basic options, one being static and one being kinetic. It is well known for static averaging that if the disorder of the underlying medium is weak, differences to the ordered case appear merely in non-universal quantities. Using dynamical field theory, we show that the same holds true for kinetic averaging. For strong disorder, i.e., the medium being close to the percolation point, we employ a field theory for the nonlinear random resistor network in conjunction with a real-world interpretation of Feynman diagrams, and we calculate the scaling exponents for the shortest, the longest and the mean or average SAW to 2-loop order. In addition, we calculate to 2-loop order the entire family of multifractal exponents that governs the moments of the the statistical weights of the elementary constituents (bonds or sites of the underlying fractal cluster) contributing to the SAWs. Our RG analysis reveals that kinetic averaging leads to renormalizability whereas static averaging does not, and hence, we argue that the latter does not lead to a well-defined scaling limit. We discuss the possible implications of this finding for experiments and numerical simulations which have produced wide-spread results for the exponent of the average SAW. To corroborate our results, we also study the well-known Meir-Harris model for SAWs on percolation clusters. We demonstrate that this model leads back to 2-loop order to the renormalizable real world formulation with kinetic averaging if the replica limit is consistently performed at the first possible instant of the calculation.
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