Hyperacceleration in a Stochastic Fermi-Ulam Model

Abstract

Fermi acceleration in a Fermi-Ulam model, consisting of an ensemble of particles bouncing between two, infinitely heavy, stochastically oscillating hard walls, is investigated. It is shown that the widely used approximation, neglecting the displacement of the walls (static wall approximation), leads to a systematic underestimation of particle acceleration. An improved approximative map is introduced, which takes into account the effect of the wall displacement, and in addition allows the analytical estimation of the long term behavior of the particle mean velocity as well as the corresponding probability distribution, in complete agreement with the numerical results of the exact dynamics. This effect accounting for the increased particle acceleration -Fermi hyperacceleration-is also present in higher dimensional systems, such as the driven Lorentz gas. In 1949 Fermi [1] proposed an acceleration mechanism of cosmic ray particles interacting with a time dependent magnetic field (for a review see [2]). Ever since, this has been a subject of intense study in a broad range of systems in various areas of physics, including astrophysics [3,4,5], plasma physics [6,7], atom optics [8,9] and has even been used for the interpretation of experimental results in atomic physics [10]. Furthermore, when the mechanism is linked to higher dimensional timedependent billiards, such as a time-dependent variant of the classic Lorentz Gas, it has profound implications on statistical and solid state physics [11]. Several modifications of the original model have been suggested, one of which is the well-known Fermi-Ulam model (FUM) [12,13,14] which describes the bouncing of a ball between an oscillating and a fixed wall. FUM and its variants have been the subject of extensive theoretical (see Ref.[13] and references therein) and experimental [15,16,17] studies as they are simple to conceive but hard to understand in that their behavior is quite complicated. A standard simplification [13] widely used in the literature, the static wall approximation (SWA), ignores the displacement of the moving wall but retains the time dependence in the momentum exchange between particle and wall at the instant of collision as if the wall were oscillating. The SWA speeds up time-consuming numerical simulations and allows semi-analytical treatments as well as a deeper understanding of the system [13,18,19,20,21]. However, as shown by Einstein in his treatment of the Brownian random walk [22], taking account of the full phase space trajectory (instead of the momentum component only) is essential for the correct description of diffusion processes. More recently, in the context of diffusion in the deterministic FUM, Lieberman et al have shown that one has to employ both canonical conjugate variables (position and momentum) in order to obtain the correct momentum distribution in the asymptotic steady state [20]. The present work shows that even in the absence of an asymptotic steady state the diffusion in velocity space is deeply affected by the location of th...

Fermi acceleration and scaling properties of a time dependent oval billiard

“…For a static boundary, after the collision, it is assumed that the particle reflects specularly ͑the angle of incidence is equal to the angle of reflection͒ with an instantaneously change in the momentum component orthogonal to the boundary. For the case of D =1 ͑one-dimensional case͒ there are many results concerning the description of Fermi acceleration and the three basic models are ͑i͒ Fermi-Ulam model, [2][3][4][5] ͑ii͒ the bouncer model, [6][7][8][9][10] and ͑iii͒ the hybrid Fermi-Ulam-bouncer model. [11][12][13] Case ͑i͒ consists of a classical particle of mass m, which is confined to bounce between two walls where one of them is fixed and the other one is periodically moving.…”

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

“…[2][3][4][5] One of the most studied versions of the problem is the onedimensional Fermi-Ulam model (FUM). [6][7][8][9][10] The model consists basically of a classical particle confined and bouncing between two rigid walls, one of them is assumed to be fixed and the other one moves according to a periodic function. For such a system, it is known that the phase space, in the absence of dissipation, is mixed, in the sense that depending on the combinations of control parameters and initial conditions, Kolmogorov-Arnold-Moser (KAM) islands, invariant spanning curves and chaotic seas are all observed.…”

Nonequilibrium fluctuations in a driven stochastic Lorentz gas

“…Such a ballistic-to-diffusive scenario, not intuitive in the presence of an accelerating field [19,22], is consistent with the autocorrelation of the particle's velocity which, up to numerical precision, decays as a single exponential.…”

“…Stochastic Lorentz models have been previously studied, focusing on transport properties, e.g. on normal or anomalous diffusion, for instance in [18][19][20][21][22][23]. Our model is based on the following ingredients: 1) the presence of an external field E accelerating the probe particle, 2) scatterers of finite mass which are randomly and uniformly distributed in space and move with random velocities as extracted from a thermal bath at temperature T (it is therefore more reasonable to call them "bath particles"), 3) collisions which can also be inelastic (the system always reaches a stationary state), 4) a uniform collision probability which is inspired from the so-called Maxwell-molecules models [24], and which helps in simplifying analytical calculations.…”