Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNoethers Theorem in Statistical Mechanics
94
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Noethers calculus of invariant variations yields exact identities from functional symmetries. The standard application to an action integral allows to identify conservation laws. Here we rather consider generating functionals, such as the free energy and the power functional, for equilibrium and driven many-body systems. Translational and rotational symmetry operations yield mechanical laws. These global identities express vanishing of total internal and total external forces and torques. We show that functional differentiation then leads to hierarchies of local sum rules that interrelate density correlators as well as static and time direct correlation functions, including memory. For anisotropic particles, orbital and spin motion become systematically coupled. The theory allows us to shed new light on the spatio-temporal coupling of correlations in complex systems. As applications we consider active Brownian particles, where the theory clarifies the role of interfacial forces in motility-induced phase separation. For active sedimentation, the center-of-mass motion is constrained by an internal Noether sum rule.
rate research
Read More
Non-autonomous non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles over the time axis R. Hamiltonian mechanics herewith can be reformulated as particular Lagrangian theory on a momentum phase space. This facts enable one to apply Noethers first theorem both to Lagrangian and Hamiltonian mechanics. By virtue of Noethers first theorem, any symmetry defines a symmetry current which is an integral of motion in Lagrangian and Hamiltonian mechanics. The converse is not true in Lagrangian mechanics where integrals of motion need not come from symmetries. We show that, in Hamiltonian mechanics, any integral of motion is a symmetry current. In particular, an energy function relative to a reference frame is a symmetry current along a connection on a configuration bundle which is this reference frame. An example of the global Kepler problem is analyzed in detail.
We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability $p$ or disappears with probability $1!-!p$. It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter $alpha$, which also determines the fragmentation rate. For a fractal dimension $d_f$, we find that the $d_f$-th moment $M_{d_f}$ is a conserved quantity, independent of $p$ and $alpha$. We use the idea of data collapse -- a consequence of dynamical scaling symmetry -- to demonstrate that the system exhibits self-similarity. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the fragmentation equation as the continuity equation of a Euclidean quantum-mechanical system. Surprisingly, the Noether charge corresponding to dynamical scaling is trivial, while $M_{d_f}$ relates to a purely mathematical symmetry: quantum-mechanical phase rotation in Euclidean time.
We show that near a second order phase transition in a two-component elastic medium of size L in two dimensions, where the local elastic deformation-order parameter couplings can break the inversion symmetry of the order parameter, the elastic modulii diverges with the variance of the local displacement fluctuations scaling as $[ln(L/a_0)]^{2/3}$ and the local displacement correlation function scaling as $[ln(r/a_0)]^{2/3}$ for weak inversion-asymmetryThe elastic constants can also vanish for system size exceeding a non-universal value, making the system unstable for strong asymmetry, where a 0 is a small-scale cut-off. We show that the elastic deformation-order parameter couplings can make the phase transition first order, when the elastic modulii do not diverge, but shows a jump proportional to the jump in the order parameter, across the transition temperature. For a bulk system, the elastic stiffness does not diverge for weak asymmetry, but can vanish across a second order transition giving instability for strong asymmetry, or displays jumps across a first order transition. In-vitro experiments on binary fluids embedded in a polymerized network, magnetic colloidal crystals or magnetic crystals could test these predictions.
We consider self-propelled particles undergoing run-and-tumble dynamics (as exhibited by E. coli) in one dimension. Building on previous analyses at drift-diffusion level for the one-particle density, we add both interactions and noise, enabling discussion of domain formation by self-trapping, and other collective phenomena. Mapping onto detailed-balance systems is possible in certain cases.
We study the statistical mechanics of double-stranded semi-flexible polymers using both analytical techniques and simulation. We find a transition at some finite temperature, from a type of short range order to a fundamentally different sort of short range order. In the high temperature regime, the 2-point correlation functions of the object are identical to worm-like chains, while in the low temperature regime they are different due to a twist structure. In the low temperature phase, the polymers develop a kink-rod structure which could clarify some recent puzzling experiments on actin.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
