Marco Bianucci

Bell-shaped curves of the enzyme activity in reverse micelles: A simplified model for hydrolytic reactions

Abstract A general characteristic of enzymes entrapped in reverse micelles (or water-in-oil microemulsions) is the bell-shape dependence of the reaction rate constant with increasing water content in the system. A simple model for a possible explanation of this bell-shape phenomenon in the particular case of hydrolytic reactions is presented. The model is based mainly on the diffusion theory and utilizes a minimum of adjustable parameters.

Nonlinear and non‐Markovian fluctuation–dissipation processes: A Fokker–Planck treatment

This paper is devoted to the derivation of the Fokker–Planck equation in a case where the external potential acting on the system and the coupling between system and bath are not harmonic. This problem has already the subject of many preceding investigations, which left open, however, the problem of deriving the Fokker–Planck equation with no assumption but the Born approximation. Within the present treatment the problem of the derivation of the Fokker–Planck equation is solved with no limitation on the bath time scale.

Beyond the linear approximations of the conventional approaches to the theory of chemical relaxation

The nonlinear coupling between the reacting system and its molecular bath results in a generalized Langevin equation with a memory kernel which is nonstationary as well as dependent on the reaction coordinate. In a preceding paper by Grigolini [J. Chem. Phys. 89, 4300 (1988)] a theory was developed to determine the reaction rate of a physical system characterized by a nonlinear interaction between system and bath. It is here shown that the local linearization adopted in that paper extends to this nonlinear condition the linear theory of Grote and Hynes, disregards also nonlinear effects, which does not conflict with the conservation of the Smoluchowski structure necessary to apply the standard first passage time approach. Here a clear distinction is made between the second‐order local linearization (SOLL) and the infinite‐order local linearization (IOLL). When deriving the Kramers equation from a microscopic description, it is possible to go beyond the SOLL approximation without contravening the basic req...

Brownian motion generated by a two-dimensional mapping

Abstract A completely deterministic derivation of the Fokker-Planck equation based on a simple 2D map is illustrated. Both friction and diffusion are derived from the properties of the chaotic booster, i.e. the map, with no ad hoc assumptions. Diffusion is shown to depend on the fact that chaos implies a sensitive dependence on initial conditions. More remarkably, friction is derived from a linear response approach, distinct from the conventional one by Kubo and taking advantage of the stability properties of the invariant distribution.

Nonconventional fluctuation dissipation process in non-Hamiltonian dynamical systems

Here, we introduce a statistical approach derived from dynamics, for the study of the geophysical fluid dynamics phenomena characterized by a weak interaction among the variables of interest and the rest of the system. The approach is reminiscent of the one developed some years ago [M. Bianucci, R. Mannella, P. Grigolini and B. J. West, Phys. Rev. E 51, 3002 (1995)] to derive statistical mechanics of macroscopic variables on interest starting from Hamiltonian microscopic dynamics. However, in the present work, we are interested to generalize this approach beyond the context of the foundation of thermodynamics, in fact, we take into account the cases where the system of interest could be non-Hamiltonian (dissipative) and also the interaction with the irrelevant part can be of a more general type than Hamiltonian. As such example, we will refer to a typical case from geophysical fluid dynamics: the complex ocean–atmosphere interaction that gives rise to the El Nino Southern Oscillation (ENSO). Here, changin...

Probing microscopic chaotic dynamics by observing macroscopic transport processes

Abstract We derive the Kramers equation, namely, the Fokker-Planck equation for an oscillator, from a completely deterministic picture. The oscillator is coupled to a “booster”, i.e., a deterministic system in a fully chaotic state, wherein diffusion is derived from the sensitive dependence of chaos on initial conditions and friction is a consequence of the linear response of the booster to the action exerted on it by the oscillator. To deal with the Hamiltonian nature of the system of interest and of its coupling to the booster, we extend the earlier theoretical derivation of macroscopic transport coefficients from deterministic dynamics. We show that the frequency of the oscillator can be tuned to the microscopic frequencies of the booster without affecting the canonical nature of the “macroscopic” statistics. The theoretical predictions are supported by numerical simulations.

Large Scale Emerging Properties from Non Hamiltonian Complex Systems

The concept of “large scale” depends obviously on the phenomenon we are interested in. For example, in the field of foundation of Thermodynamics from microscopic dynamics, the spatial and time large scales are order of fraction of millimetres and microseconds, respectively, or lesser, and are defined in relation to the spatial and time scales of the microscopic systems. In large scale oceanography or global climate dynamics problems the time scales of interest are order of thousands of kilometres, for space, and many years for time, and are compared to the local and daily/monthly times scales of atmosphere and ocean dynamics. In all the cases a Zwanzig projection approach is, at least in principle, an effective tool to obtain class of universal smooth “large scale” dynamics for few degrees of freedom of interest, starting from the complex dynamics of the whole (usually many degrees of freedom) system. The projection approach leads to a very complex calculus with differential operators, that is drastically simplified when the basic dynamics of the system of interest is Hamiltonian, as it happens in Foundation of Thermodynamics problems. However, in geophysical Fluid Dynamics, Biology, and in most of the physical problems the building block fundamental equations of motions have a non Hamiltonian structure. Thus, to continue to apply the useful projection approach also in these cases, we exploit the generalization of the Hamiltonian formalism given by the Lie algebra of dissipative differential operators. In this way, we are able to analytically deal with the series of the differential operators stemming from the projection approach applied to these general cases. Then we shall apply this formalism to obtain some relevant results concerning the statistical properties of the El Nino Southern Oscillation (ENSO).

THE LINEAR RESPONSE APPROACH TO THE FOKKER-PLANCK EQUATION I: THEORY

We show that a Brownian motion can be derived for a particle coupled to a bath with a finite number of chaotic degrees of freedom, using no thermodynamics assumptions. We point out that although diffusion is easily derived in terms of the unperturbed bath via the central limit theorem, friction is obtained only when the coupling between particle of interest and bath is considered. The only assumption made is that the bath should respond linearly (in statistical sense) to a weak external perturbation. We prove that the calculation of the friction can be traced back to the bath susceptibility.

THE LINEAR-RESPONSE APPROACH TO THE FOKKER-PLANCK EQUATION .3. A DETERMINISTIC AND CHAOTIC BOOSTER

We apply the theory developed in I1 to two deterministic nonlinear systems with few degrees of freedom (mappings). The first mapping considered is known to exactly satisfy the prescriptions laid down in I to obtain the Fokker-Planck equation associated to Brownian motion. For the second mapping, due to its more complex form, it is not possible to prove analytically that it too satisfies the prescriptions of I; however, we show numerically that this fact is plausible, at least in the chaotic regime. For both cases we show that indeed Brownian motion for the variable of interest w arises. We conclude arguing that the theory developped in I is generally applicable to systems for which the “thermal bath” is in a fully chaotic state.

THE LINEAR-RESPONSE APPROACH TO THE FOKKER-PLANCK EQUATION .2. A NONLINEAR STOCHASTIC BOOSTER

It has been pointed out in I1 that the friction resulting from the interaction between a Brownian particle and a nonlinear bath cannot in general be evaluated by a second-order perturbation treatment with the conventional Liouville or Liouville-like approach. It has also been shown that a convenient way of proceeding is to express the dynamics of the bath within a master equation approach, which makes it possible to apply a linear response treatment (LRT) also in the case of a nonlinear bath. We check the internal consistency of this theory using as a bath for the velocity w of the Brownian particle a doorway variable ξ, which executes random jumps among some discrete values. In this specific condition the explicit expression for the master equation driving the motion of the bath is easily derived. It is also relatively easy to check the predictions of the theory with computer calculations. The internal consistency of the theory is proved and the agreement with the numerical calculation is shown to be remarkably good in the parameter region where the LRT approach is expected to hold.