arXiv subjects

Daniele Venturi

Publications and source records attributed to Daniele Venturi.

39 records · Page 3Linked to original sources

arXiv2017

The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations

The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations) and statistical physics (equations for generating functionals and effective Fokker-Planck equations). However, no effective numerical method has yet been developed to compute their solution. The purpose of this report is to fill this gap, and provide a new perspective on the problem of numerical approximation of nonlinear functionals and functional differential equations.

math.NA↗

arXiv2017

Duality and Conditional Expectations in the Nakajima-Mori-Zwanzig Formulation

We develop a new operator algebraic formulation of the Nakajima-Mori-Zwanzig (NMZ) method of projections. The new theory is built upon rigorous mathematical foundations, and it can be applied to both classical and quantum systems. We show that a duality principle between the NMZ formulation in the space of observables and in the state space can be established, analogous to the Heisenberg and Schrödinger pictures in quantum mechanics. Based on this duality we prove that, under natural assumptions, the projection operators appearing in the NMZ equation must be conditional expectations. The proposed formulation is illustrated in various examples.

math-ph↗

arXiv2013

Conjugate flow action functionals

We present a new general method to construct an action functional for a non-potential field theory. The key idea relies on representing the governing equations of the theory relative to a diffeomorphic flow of curvilinear coordinates which is assumed to be functionally dependent on the solution field. Such flow, which will be called the conjugate flow of the theory, evolves in space and time similarly to a physical fluid flow of classical mechanics and it can be selected in order to symmetrize the Gateaux derivative of the field equations with respect to suitable local bilinear forms. This is equivalent to requiring that the governing equations of the field theory can be derived from a principle of stationary action on a Lie group manifold. By using a general operator framework, we obtain the determining equations of such manifold and the corresponding conjugate flow action functional. In particular, we study scalar and vector field theories governed by second-order nonlinear partial differential equations. The identification of transformation groups leaving the conjugate flow action functional invariant could lead to the discovery of new conservation laws in fluid dynamics and other disciplines.

math-ph↗