Scivolemaj

Abstract
In the present article we will use the formalism of Mori and the theory of the Brownian motors to maximize, for one given memory and energy barrier, the speed of an on-off ratchet.

I. Introduction
In 1827, the botanical, Robert Brown studied, initially using a microscope, the movement of particles, contained in the pollen of the plants, when placed in water. Later, it studied several other types of particles, also the inorganics, and used other watery substances. In all these cases, he observed an irregular and incessant movement of particles. Owing to Robert Brown, the movement is called brownian. Since then, many researchers, as Regnault (1858), Chr. Wiener (1863), Cantoni and Oehl (1865), Gouy(1888), F.M. Exner (1900), had started to study this movement. In April of 1905, Einstein finishes its thesis of doctorate, that 'it' treats on the particle diffusion of soluto in a diluted solution (particles of sugar in water), and in May of the same year it publishes an article on small particles in suspension in a stationary liquid. In the article, Einstein finds a relation for the coefficient of diffusion of particles and deduces, in a new way, the diffusion equation, anticipating the relation of Chapman-Kolmogorov and the modern theories of markovian chains. In the following year, Smoluchowski publishes its article on the brownian movement, subject of its interest since 1900, and, even so it has followed a different method of the one of Einstein, he arrives at similar results. Smoluchowski, in its article, compares the theory with the experimental results gotten by Felix Exner and finds a good agreement. In 1908, Langevin presents a new boarding for the movement of particles in a fluid, in which directly uses the second law of Newton. The force that acts on the particle is divided in two components: one is the force of Stokes and the other one is a random force. Both the forces have its origin in the interactions between molecules of the fluid with the brownian particle. This connection becomes explicit by means of a relation that was known as Fluctuation-Dissipation theorem. The theories of Einstein and Langevin had been confirmed experimentally by Jean Perrin in careful experiences. The theory of Langevin got the same results found by Einstein and opened way for extensions of the theory of the diffusion. Years later, Mori, using the equation of Liouville in the formalism of the Hilbert space, deduces a generalization of the Langevin equation. This equation, besides describing more precisely the non markovian physical systems, helped to base theoretically the mechanics statistics out of the balance that previously was boarded in a fenomenologic way. In the present article we use the equations of Langevin as 'a' base for our research on brownian motors. Brownian motors are, for our purposes, systems out of the balance, which move particles in a preferential direction by means of the rectification of the thermal fluctuations. Operating at energies just above those of the thermal bath, these motors experience large fluctuations, and their physical description must be necessarily stochastic. The theory of the Brownian motors has his roots in different fields. One of this, naturally, is the study of the brownian movement whose evolution 'caused' questions. The answers to them constitute the base on which the recent results are established. Already in 1912, Marian Smoluchowski showed, in an imaginary experiment, that a microscopic apparatus constituted by paddles, an axle, a ratchet and a pawl, subjected to a thermal bath, is incapable of rectifying the thermal fluctuations. Feynman extended the results of Smoluchowski for the case where it has two thermal baths at different temperatures and demonstrated that in this situation it is possible to rectify the fluctuations. These experiences retrace to the famous thermodynamic demon of Maxwell and they do not violate the second law of the thermodynamics. Although the theory has originated in the context of fluid mechanics, with applications in particle separation for example, nowadays we can find, also, other applications: diodes, transistors, photo liquid crystal alignment and even though the development of technology SQUID (Superconducting quantum interference device). Other branches that also study and develop theories on the brownian motors are the biochemist and the biophysics. The interest for these fields is in the possibility to understand, among others, the active intracellular substance transport, cellular division, the molecular pumps and the muscular contraction.

II. The Mori formalism and the brownian motors
The models for Brownian motors with white noise are simple, but in a certain way ingenuous, since that more realistic systems present correlation. In this way our objective will be to study the Brownian motor with correlation. For such intention we go to use the formalism of Mori. In the year of 1964, Hazime Mori, Japanese physicist, generalized the Langevin equation. Its equation was rigorously deduced from first principles and allows to describe phenomena that are not markovian. We will use the following generalized Langevin equation:

$$\frac{dv\left( t\right) }{dt}=-\int_{0}^{t}\Gamma \left( t-t_{1}\right) v\left( t_{1}\right) dt_{1}+\sqrt{\frac{k_{b}T}{m}}F_{a}(t)+F_{e}+F_{Pot}(x,t) \label{ELG1}$$

where

$$F_{a}(t)$$ is a random force, $$F_{e}$$ is a constant force and $$F_{Pot}(x,t)$$ is a force that depends on x and t.

The random force satisfies the following relations:

$$\left\langle F_{a}(t)\right\rangle =0 \label{Flutuacao-dissipacao}$$ and

$$\left\langle F_{a}(t_{1})F_{a}\left( t_{2}\right) \right\rangle =\Gamma \left( t_{1}-t_{2}\right).$$

In the present article we will study one On-Off ratchet governed by the equation:

$$\ddot{x}\left( t\right) =-\int_{0}^{t}\Gamma \left( t-t_{1}\right) \dot{x}% \left( t_{1}\right) dt_{1}+\sqrt{\frac{k_{b}T}{m}}% F_{a}(t)+F_{e}+F_{Pot}(x)(1+f(t)). \label{motmemory}$$

$$f(t)$$ can assume two values 0 or -1 - potential on or off respectively.

The sawtooth potential posses the form of the following figure:

The sawtooth potential. :Potencial.jpg

One on-off ratchet, basically, repeats 3 stages - see figure:

1 - First the particles are kept imprisoned in the first well by potential U for a time $$t_{on}$$;

2- Second the potential is off during a time $$t_{off}$$ and the particles spread out with a gaussian probability distribution as in the figure below;

3- the potential is turned on again and most of particles is captured in the front well.

The formal solution for the equation (blabla) when potential is off is:

$$x(t_{off})=x(0)+v(0)H(t_{off})+F_{e}\int% \nolimits_{0}^{t_{off}}H(t_{off}-t_{1})dt_{1}+\sqrt{\frac{k_{b}T}{m}}% \int\nolimits_{0}^{t_{off}}H(t_{off}-t_{2})F_{a}(t_{2})dt_{2} \label{Xtoff}$$

where $$\widetilde{H}(s)=\frac{1}{s^{2}+s\widetilde{\Gamma }(s)} $$ is the laplace transform of $$H(t)$$.

Fa is an integral of independent gaussian random numbers $$\phi (\omega )$$:

$$F_{a}(t)=\int_{0}^{\infty }\sqrt{\frac{2\overline{\Gamma }(\omega )}{\pi }}% \cos [\omega t+\phi (\omega )]d\omega =\int_{0}^{\infty }\sqrt{\frac{2% \overline{\Gamma }(\omega )}{\pi }}[\cos (\phi (\omega ))\cos (\omega t)-\sin (\phi (\omega ))\sin (\omega t)]d\omega $$

where

$$\left\langle \phi (\omega )\phi (\omega ^{/})\right\rangle =\delta \left( \omega -\omega ^{/}\right),$$

$$\left\langle \cos (\phi (\omega ))\cos (\phi (\omega ^{/})\right\rangle =\left\langle \sin (\phi (\omega ))\sin (\phi (\omega ^{/})\right\rangle =% \frac{1}{2}\delta \left( \omega -\omega ^{/}\right)$$

and

$$\left\langle \cos (\phi (\omega ))\sin (\phi (\omega ^{/})\right\rangle =0.$$

Consequently $$x(t_{off})$$ will have a gaussian distribution:

$$P(x,t_{off})=\frac{1}{\sqrt{2\pi d}}\exp \left[ -\frac{1}{2d}\left( x-\left\langle x(t_{off})\right\rangle \right) ^{2}\right], $$

with

$$\left\langle x(t_{off})\right\rangle =\left\langle x(0)\right\rangle +\left\langle v(0)\right\rangle H(t_{off})+F_{e}\int\nolimits_{0}^{t_{off}}H(t_{off}-t_{1})dt_{1}, \label{Xtoff}$$

and

$$d=2\int_{0}^{t_{off}}H(t_{1})dt_{1}-(H(t_{off}))^{2}.$$

Conclusions
Studi Brownian motors!

acknowledgments
Beautiful womans.