10 research outputs found
Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers
Some results on the ordered statistics of eigenvalues for one-dimensional
random Schr\"odinger Hamiltonians are reviewed. In the case of supersymmetric
quantum mechanics with disorder, the existence of low energy delocalized states
induces eigenvalue correlations and makes the ordered statistics problem
nontrivial. The resulting distributions are used to analyze the problem of
classical diffusion in a random force field (Sinai problem) in the presence of
weakly concentrated absorbers. It is shown that the slowly decaying averaged
return probability of the Sinai problem, \mean{P(x,t|x,0)}\sim \ln^{-2}t, is
converted into a power law decay, \mean{P(x,t|x,0)}\sim t^{-\sqrt{2\rho/g}},
where is the strength of the random force field and the density of
absorbers.Comment: 10 pages ; LaTeX ; 4 pdf figures ; Proceedings of the meeting
"Fundations and Applications of non-equilibrium statistical mechanics",
Nordita, Stockholm, october 2011 ; v2: appendix added ; v3: figure 2.left
adde
One-dimensional classical diffusion in a random force field with weakly concentrated absorbers
A one-dimensional model of classical diffusion in a random force field with a
weak concentration of absorbers is studied. The force field is taken as
a Gaussian white noise with \mean{\phi(x)}=0 and \mean{\phi(x)\phi(x')}=g
\delta(x-x'). Our analysis relies on the relation between the Fokker-Planck
operator and a quantum Hamiltonian in which absorption leads to breaking of
supersymmetry. Using a Lifshits argument, it is shown that the average return
probability is a power law \smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2\rho/g}} (to be
compared with the usual Lifshits exponential decay in
the absence of the random force field). The localisation properties of the
underlying quantum Hamiltonian are discussed as well.Comment: 6 pages, LaTeX, 5 eps figure
Sinai model in presence of dilute absorbers
We study the Sinai model for the diffusion of a particle in a one dimension
random potential in presence of a small concentration of perfect
absorbers using the asymptotically exact real space renormalization method. We
compute the survival probability, the averaged diffusion front and return
probability, the two particle meeting probability, the distribution of total
distance traveled before absorption and the averaged Green's function of the
associated Schrodinger operator. Our work confirms some recent results of
Texier and Hagendorf obtained by Dyson-Schmidt methods, and extends them to
other observables and in presence of a drift. In particular the power law
density of states is found to hold in all cases. Irrespective of the drift, the
asymptotic rescaled diffusion front of surviving particles is found to be a
symmetric step distribution, uniform for , where
is a new, survival length scale ( in the absence of
drift). Survival outside this sharp region is found to decay with a larger
exponent, continuously varying with the rescaled distance . A simple
physical picture based on a saddle point is given, and universality is
discussed.Comment: 21 pages, 2 figure
Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
The concept of Lyapunov exponent has long occupied a central place in the
theory of Anderson localisation; its interest in this particular context is
that it provides a reasonable measure of the localisation length. The Lyapunov
exponent also features prominently in the theory of products of random matrices
pioneered by Furstenberg. After a brief historical survey, we describe some
recent work that exploits the close connections between these topics. We review
the known solvable cases of disordered quantum mechanics involving random point
scatterers and discuss a new solvable case. Finally, we point out some
limitations of the Lyapunov exponent as a means of studying localisation
properties.Comment: LaTeX, 23 pages, 3 pdf figures ; review for a special issue on
"Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor change
Breaking supersymmetry in a one-dimensional random Hamiltonian
The one-dimensional supersymmetric random Hamiltonian
, where is a Gaussian white
noise of zero mean and variance , presents particular spectral and
localization properties at low energy: a Dyson singularity in the integrated
density of states (IDoS) and a delocalization transition
related to the behaviour of the Lyapunov exponent (inverse localization length)
vanishing like as . We study how this picture
is affected by breaking supersymmetry with a scalar random potential:
where is a Gaussian white noise of variance .
In the limit , a fraction of states
migrate to the negative spectrum and the
Lyapunov exponent reaches a finite value at
E=0. Exponential (Lifshits) tail of the IDoS for is studied in
detail and is shown to involve a competition between the two noises and
whatever the larger is. This analysis relies on analytic results for
and obtained by two different methods: a stochastic method and the
replica method. The problem of extreme value statistics of eigenvalues is also
considered (distribution of the n-th excited state energy). The results are
analyzed in the context of classical diffusion in a random force field in the
presence of random annihilation/creation local rates.Comment: 33 pages, LaTeX, 13 eps figures ; 2nd version : refs. adde