اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدProblems with the definition of renormalized Hamiltonians for momentum-space renormalization transformations
79
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
For classical lattice systems with finite (Ising) spins, we show that the implementation of momentum-space renormalization at the level of Hamiltonians runs into the same type of difficulties as found for real-space transformations: Renormalized Hamiltonians are ill-defined in certain regions of the phase diagram.
قيم البحث
اقرأ أيضاً
We introduce a general method for optimizing real-space renormalization-group transformations to study the critical properties of a classical system. The scheme is based on minimizing the Kullback-Leibler divergence between the distribution of the sy
stem and the normalized normalizing factor of the transformation parametrized by a restricted Boltzmann machine. We compute the thermal critical exponent of the two-dimensional Ising model using the trained optimal projector and obtain a very accurate thermal critical exponent $y_t=1.0001(11)$ after the first step of the transformation.
Renormalization group calculations are used to give exact solutions for rigidity percolation on hierarchical lattices. Algebraic scaling transformations for a simple example in two dimensions produce a transition of second order, with an unstable cri
tical point and associated scaling laws. Values are provided for the order parameter exponent $beta = 0.0775$ associated with the spanning rigid cluster and also for $d u = 3.533$ which is associated with an anomalous lattice dimension $d$ and the divergence in the correlation length near the transition. In addition we argue that the number of floppy modes $F$ plays the role of a free energy and hence find the exponent $alpha$ and establish hyperscaling. The exact analytical procedures demonstrated on the chosen example readily generalize to wider classes of hierarchical lattice.
We demonstrate the power of a recently-proposed approximation scheme for the non-perturbative renormalization group that gives access to correlation functions over their full momentum range. We solve numerically the leading-order flow equations obtai
ned within this scheme, and compute the two-point functions of the O(N) theories at criticality, in two and three dimensions. Excellent results are obtained for both universal and non-universal quantities at modest numerical cost.
The stochastic solution with Gaussian stationary increments is establihsed for the symmetric space-time fractional diffusion equation when $0 < beta < alpha le 2$, where $0 < beta le 1$ and $0 < alpha le 2$ are the fractional derivation orders in tim
e and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuos time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal Levy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure. Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.
We develop a theoretical approach to ``spontaneous stochasticity in classical dynamical systems that are nearly singular and weakly perturbed by noise. This phenomenon is associated to a breakdown in uniqueness of solutions for fixed initial data and
underlies many fundamental effects of turbulence (unpredictability, anomalous dissipation, enhanced mixing). Based upon analogy with statistical-mechanical critical points at zero temperature, we elaborate a renormalization group (RG) theory that determines the universal statistics obtained for sufficiently long times after the precise initial data are ``forgotten. We apply our RG method to solve exactly the ``minimal model of spontaneous stochasticity given by a 1D singular ODE. Generalizing prior results for the infinite-Reynolds limit of our model, we obtain the RG fixed points that characterize the spontaneous statistics in the near-singular, weak-noise limit, determine the exact domain of attraction of each fixed point, and derive the universal approach to the fixed points as a singular large-deviations scaling, distinct from that obtained by the standard saddle-point approximation to stochastic path-integrals in the zero-noise limit. We present also numerical simulation results that verify our analytical predictions, propose possible experimental realizations of the ``minimal model, and discuss more generally current empirical evidence for ubiquitous spontaneous stochasticity in Nature. Our RG method can be applied to more complex, realistic systems and some future applications are briefly outlined.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
