# Results

Some results I have contributed to.

#### Unraveling of Completely Bounded Divisible Dynamical Maps

$$ \boldsymbol{\rho}_{t} = \operatorname{E} \Big{(}\mu_{t}\boldsymbol{\psi}_{t}\boldsymbol{\psi}_{t}^{\dagger}\Big{)} $$
We proved that any completely bounded divisible dynamical maps admis a representation as a statistical average over the solution of a Markov process taking values in the Hilbert state of the open quantum system. Applications are time continuous measurement interpretation and numerically efficient Monte Carlo algorithms for quantum master equation generating a semi-group.

#### Refined Second Law for Markov Processes

$$ E\,\geq\,\mathrm{K_{B}}\,T\,\ln 2 $$
In a series of distinct collaborations, we proved that thermodynamic transitions in non-equilibrium thermodynamics described by Markov processes (overdamped, underdamped with arbitrary or kinetic plus potential Hamiltonian, pure jump ) always occur with average entropy production
bounded away from zero. Optimal control problems are notoriously sensitive to the class of admissible protocols. In all the cases above, optimal protocol are smooth and non-equilibrium: the current velocity at the end of the control horizon is non-vanishing.

#### Multiscaling in the Kraichnan Model of Passive Advection

$$ \mathcal{S}_{2\,n}(\boldsymbol{x};m)=\operatorname{E}|\theta(\boldsymbol{x};m)-\theta(\boldsymbol{0};m)|^{2\,n}\propto \|\boldsymbol{x}\|^{\zeta_{2\,n}(\xi)} $$
Kraichnan's model of passive advection is to-date the only PDE model of turbulence where intermittency and multiscaling can be proven in a mathematically controlled fashion. The result was first independently derived in the diffusion , large dimension , and laminar limits and numerically also by us here.

#### Functional Determinants in the Stationary Action Approximation

$$ \frac{\mathrm{Det}O_{\mathcal{A},t_{\mathfrak{0}},\mathfrak{t_{1}}}^{(u_{\mathfrak{1}})} }{\mathrm{Det}O_{\mathcal{A},t_{\mathfrak{0}},\mathfrak{t_{1}}}^{(u_{\mathfrak{1}})}}=\frac{\mathrm{det}(A_{\mathfrak{0}}+A_{\mathfrak{1}}\Phi_{*\, t_{\mathfrak{0}},\mathfrak{t_{1}}}^{(u_{\mathfrak{1}})})}{\mathrm{det}(A_{\mathfrak{0}}+A_{\mathfrak{1}}\Phi_{*\, t_{\mathfrak{0}},\mathfrak{t_{1}}}^{(u_{\mathfrak{1}})})} $$
A beautiful result by Robin Forman yields a powerful expression of functional determinants occurring in elliptic boundary value problems.

#### The Concept of "Strong Anomalous Diffusion"

$$ \operatorname{E}|\xi_{t}-\operatorname{E}\xi_{t}|^{q}\propto t^{q\,\nu(q)} $$
In Castiglione et al (1999) we inquired the mechanisms
underlying genuine multiscaling in diffusion processes. Strong anomalous diffusion appears in fine-tuned parametric regions of chaotic dynamical systems.