Quasipotentials in the nonequilibrium stationary states or a method to get explicit solutions of Hamilton–Jacobi equations

We assume that a system at a mesoscopic scale is described by a field ϕ(x, t) that evolves by a Langevin equation with a white noise whose intensity is controlled by a parameter 1 / Ω . The system stationary state distribution in the small noise limit (Ω → ∞) is of the form P st [ϕ] ≃ exp(−ΩV 0[ϕ]), where V 0[ϕ] is called the quasipotential. V 0 is the unknown of a Hamilton–Jacobi equation. Therefore, V 0 can be written as an action computed along a path that is the solution from Hamilton’s equation that typically cannot be solved explicitly. This paper presents a theoretical scheme that builds a suitable canonical transformation that permits us to do such integration by deforming the original path into a straight line and including some weights along with it. We get the functional form of such weights through conditions on the existence and structure of the canonical transformation. We apply the scheme to get the quasipotential algebraically for several one-dimensional nonequilibrium models as the diffusive and reaction–diffusion systems.

Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation—not just a formal power series—bringing the Hamiltonian into its Birkhoff normal form.

Adaptive technology for constructing the kinetic equations of reduction reactions under conditions of a priori uncertainty

The object of research is the process of oxide reduction in a reaction system of mass m due to the reaction on a contact surface with an area of S. An adaptive technology is proposed that allows one to construct the kinetic equation of the process in which the oxide is reduced from the initial product under conditions of a priori uncertainty. A priori uncertainty regarding the behavior of a physicochemical system is understood as the fact that the following information is not available to the researcher: – about the change in the mass of the reaction system and the area of the contact surface; – about the rate of accumulation of the finished product; – about the time of withdrawal of the finished product from the system. The proposed adaptive technology includes five sequential stages to eliminate a priori uncertainty. This is ensured through the use of an adaptive algorithm, which allows obtaining the maximum accuracy in estimating the output variable by selecting the optimal parameter of the adaptive algorithm, and the subsequent canonical transformation. The introduced concept "canonical transformation of the kinetic equation" has the following meaning: having received some adequate description of the kinetic equation in a Cartesian coordinate system, a transformation is carried out that allow representing the equation in a new Cartesian coordinate system in such a way that its structure corresponds to the canonical form. The basic postulate of chemical kinetics can be such a canonical type.

A new Wilson line-based action for gluodynamics

We perform a canonical transformation of fields that brings the Yang-Mills action in the light-cone gauge to a new classical action, which does not involve any triple-gluon vertices. The lowest order vertex is the four-point MHV vertex. Higher point vertices include the MHV and $$ \overline{\mathrm{MHV}} $$ MHV ¯ vertices, that reduce to the corresponding amplitudes in the on-shell limit. In general, any n-leg vertex has 2 ≤ m ≤ n − 2 negative helicity legs. The canonical transformation of fields can be compactly expressed in terms of path-ordered exponentials of fields and their functional derivative. We apply the new action to compute several tree-level amplitudes, up to 8-point NNMHV amplitude, and find agreement with the standard methods. The absence of triple-gluon vertices results in fewer diagrams required to compute amplitudes, when compared to the CSW method and, obviously, considerably fewer than in the standard Yang-Mills action.

Dirac’s Formalism for Time-Dependent Hamiltonian Systems in the Extended Phase Space

Dirac’s formalism for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. We show that the Lewis invariant is a reparametrization invariant, and we calculate the Feynman propagator using the extended phase space description. We show that the Feynman propagator’s quantum phase is given by the boundary term of the canonical transformation of the extended phase space. We propose a new canonical transformation within the extended phase space that leads to a Lewis invariant generalization, and we sketch some possible applications.

Fixing the non-relativistic expansion of the 1PM potential

We obtain a first order post-Minkowskian two-body effective potential whose post-Newtonian expansion directly reproduces the Einstein-Infeld-Hoffmann potential. Post-Minkowskian potentials can be extracted from on-shell scattering amplitudes in a quantum field theory of scalar matter coupled to gravity. Previously, such potentials did not reproduce the Einstein-Infeld-Hoffmann potential without employing a suitable canonical transformation. In this work, we resolve this issue by obtaining a new expression for the first-order post-Minkowskian potential. This is accomplished by exploiting the reference frame dependence that arises in the scattering amplitude computation. Finally, as a check on our result, we demonstrate that our new potential gives the correct scattering angle.

FEATURES OF FINDING OPTIMAL SOLUTIONS IN NETWORK PLANNING

The object of research is a test network diagram, in relation to which the task of minimizing the objective function qmax/qmin→min is posed, which requires maximizing the uniformity of the workload of personnel when implementing an arbitrary project using network planning. The formulation of the optimization problem, therefore, assumed finding such times of the beginning of the execution of operations, taken as input variables, in order to ensure the minimum value of the ratio of the peak workload of personnel to the minimum workload. The procedure for studying the response surface proposed in the framework of RSM is described in relation to the problem of optimizing network diagrams. A feature of this procedure is the study of the response surface by a combination of two methods – canonical transformation and ridge analysis. This combination of methods for studying the response surface allows to see the difference between optimal solutions in the sense of "extreme" and in the sense of "best". For the considered test network diagram, the results of the canonical transformation showed the position on the response surface of the extrema in the form of maxima, which is unacceptable for the chosen criterion for minimizing the objective function qmax/qmin→min. It is shown that the direction of movement towards the best solutions with respect to minimizing the value of the objective function is determined on the basis of a parametric description of the objective function and the restrictions imposed by the experiment planning area. A procedure for constructing nomograms of optimal solutions is proposed, which allows, after its implementation, to purposefully choose the best solutions based on the real network diagrams of your project