Ballisticity and Einstein relation in 1d Mott variable range hopping Alessandra Faggionato Department of Mathematics University La Sapienza Joint work with N. Gantert and M. Salvi Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Physical motivations Phonon–assisted electron transport in disordered solids in the regime of strong Anderson localization (e.g. doped semiconductors) • : impurities located at x i E i : energy mark associated to x i { x i } and { E i } are random Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Physical motivations • Electrons are localized around impurities • E i = energy of electron around x i • η ∈ { 0 , 1 } N � 1 there is electron around x i • η i = 0 otherwise Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Simple exclusion process with site disorder • Probability rate for an electron to hop from x i to x j : exp {−| x i − x j | − β { E j − E i } + } e − β ( Ei − λ ) • µ λ : reversible product probability, µ λ ( η i ) = 1+ e − β ( Ei − λ ) • Interesting regime: β → ∞ • Independent particle approximation : probability rate for a jump x i � x j µ λ ( η i = 1 , η j = 0) exp {−| x i − x j | − β { E j − E i } + } ≈ exp {−| x i − x j | − β 2 ( | E i − λ | + | E j − λ | + | E i − E j | ) } Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
• A. Miller, E. Abrahams, Impurity Conduction at Low Concentrations . Phys. Rev. 120 , 745-755 (1960) • V. Ambegoakar, B. Halperin, J.S. Langer, Hopping conductivity in disordered systems . Phys. Rev. B 4 , 2612–2620 (1971). Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
• { x i } = Z d , nearest–neighbor jumps • Hydrodynamic limit: F., Martinelli (PTRF 2003); Quastel (AP 2006) • ∂ t m = ∇ ( D ( m ) ∇ m ) • Quastel (AP 2006): lim m → 0 D ( m ) = D (0), D (0) diffusion matrix random walk with jump rates obtained by a similar procedure Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Continuous–time random walk X ξ t Environment: ξ = ( { x i } , { E i } ) • X ξ t ∈ { x i } , • X ξ 0 = 0, • Given x i � = x j , probability rate for a jump x i � x j is r x i , x j ( ξ ) = exp {−| x i − x j | − β ( | E i | + | E j | + | E i − E j | ) } Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Variable range hopping r x i , x j ( ξ ) = exp {−| x i − x j | − β ( | E i | + | E j | + | E i − E j | ) } • Low temperature regime: β → ∞ . • Long jumps can become convenient if energetically nice Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Mott–Efros–Shklovskii law In d ≥ 2 the contribution of long jumps dominates as β → ∞ • For genuinely nearest neighbor random walk diffusion matrix D ( β ) = O ( e − cβ ) • Mott–Efros–Shklovskii law (for isotropic environment): α +1 � � D ( β ) ∼ exp − c β 1 α +1+ d if P ( E i ∈ [ E, E + dE )) = c | E | α dE , α ≥ 0. • Rigorous lower/upper bounds: A.F. D.Spehner, H. Schulz–Baldes CMP (2006); A.F., P.Mathieu CMP (2008) • M-E-S law concerns conductivity σ ( β ). If Einstein relation is not violated, then σ ( β ) = βD ( β ) Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Diffusive/Subdiffusive behavior E E E E E 2 −2 −1 0 1 x −2 x 0 x x 2 x −1 1 Z Z −1 Z Z −2 0 1 =0 Theorem ( A.F., P. Caputo AAP (2009)) � e Z 0 � • If E < ∞ , then quenched invariance principle and c 1 exp {− κ 1 β } ≤ D ( β ) ≤ c 2 exp {− κ 2 β } . � e Z 0 � • If E = ∞ , then annealed invariance principle and D ( β ) = 0 . Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Einstein relation for random walks in random environment • J. Lebowitz, H. Rost (SPA 1994) • Tagged particle in a dynamical random environment with positive spectral gap: T. Komorowski, S. Olla (JSP 2005) • Reversible diffusion in random environment: Gantert, Mathieu, Piatnitski (CPAM 2012) • ... Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Biased 1d Mott random walk Joint work with N. Gantert, M. Salvi (2016) E E E E E 2 −2 −1 0 1 x −2 x 0 x x 2 x −1 1 Z Z −1 Z Z −2 0 1 =0 Take λ ∈ (0 , 1) and u ( · , · ) bounded, symmetric r λ x i ,x j ( ξ ) = exp {−| x i − x j | + λ ( x j − x i ) − u ( E i , E j ) } Biased random walk ( X ξ,λ ) t ≥ 0 is well defined. t Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Assumptions: • (A1) The sequence ( Z k , E k ) k ∈ Z is ergodic and stationary w.r.t. shifts; • (A2) The expectation E ( Z 0 ) is finite; • (A3) There exists ℓ > 0 satisfying P ( Z 0 ≥ ℓ ) = 1. Transience Proposition For P –a.a. ξ the rw X ξ,λ is transient to the right: t • lim t →∞ X ξ,λ = + ∞ a.s. t Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Ballistic/Subballistic behavior Theorem � e (1 − λ ) Z 0 � • If E < ∞ , then for P –a.a. ξ it holds X ξ,λ t lim = v ( λ ) > 0 a.s. t t →∞ e − (1+ λ ) Z − 1 +(1 − λ ) Z 0 � � • If E = ∞ , then for P –a.a. ξ it holds X ξ,λ t lim = v ( λ ) = 0 a.s. t t →∞ Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Comments � e (1 − λ ) Z 0 � � < ∞ ⇒ v ( λ ) > 0 E � e − (1+ λ ) Z − 1 +(1 − λ ) Z 0 � = ∞ ⇒ v ( λ ) = 0 E • If ( Z k ) k ∈ Z are i.i.d., or in general if � E ( Z − 1 | Z 0 ) � ∞ < ∞ , then e (1 − λ ) Z 0 � � < ∞ ⇐ ⇒ v ( λ ) > 0 E • Previous theorem holds for Y ξ,λ n = jump process of X ξ,λ t r λ xi,xj ( ξ ) xi,xk ( ξ ) probability for Y ξ,λ p λ x i ,x k ( ξ ) = to x i � x j n k r λ � Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
• Y ξ,λ n : discrete time random walk • p λ x i ,x k ( ξ ): probability to jump from x i to x k k x k p λ • ϕ λ ( ξ ) = � 0 ,x k ( ξ ) local drift Theorem e (1 − λ ) Z 0 � � Suppose that E < ∞ . The environment viewed from Y ξ,λ has an invariant ergodic distribution Q λ mutually n absolutely continuous w.r.t. P , v Y ( λ ) � � v Y ( λ ) = Q λ ϕ λ and v X ( λ ) = � � k r λ 1 / ( � 0 ,x k ) Q λ True also for λ = 0: � k r 0 ,xk d Q 0 = k r 0 ,xk ] d P reversible, v Y (0) = v X (0) = 0 E [ � Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Warning When λ = 0, λ is understood: r x i ,x j ( ξ ), p x i ,x k ( ξ ), X ξ t , Y ξ n Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Cut-off • ρ : positive integer • Consider Y ξ,λ , and suppress jumps of length larger than ρ . n • Q ( ρ ) λ : invariant ergodic distribution for the new random walk, absolutely continuous w.r.t. P . • Probabilistic representation of d Q ( ρ ) d P . λ • Q ( ρ ) weakly converges to Q λ . λ • F. Comets, S. Popov, AIHP 48 , 721–744 (2012) Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Estimates on the Radon–Nykodim derivative d Q λ d Q 0 Proposition e pZ 0 � � Suppose that for some p ≥ 2 it holds E < + ∞ . Fix λ 0 ∈ (0 , 1) . Then � d Q λ � � sup L p ( Q 0 ) < ∞ � � d Q 0 � λ ∈ (0 ,λ 0 ) Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Continuity of Q λ ( f ) at λ = 0 Theorem Suppose that E (e pZ 0 ) < ∞ for some p ≥ 2 and let q be the coniugate exponent, i.e. q satisfies 1 p + 1 q = 1 . If f ∈ L q ( Q 0 ) , then f ∈ L 1 ( Q λ ) for λ ∈ (0 , 1) and λ → 0 Q λ ( f ) = Q 0 ( f ) lim Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
∂ λ =0 Q λ ( f ) • τ x k ξ : environment translated to make x k the new origin k p 0 ,x k [ f ( τ x k ξ ) − f ( ξ )] for f ∈ L 2 ( Q 0 ) • L 0 f ( ξ ) = � • f ∈ L 2 ( Q 0 ) ∩ H − 1 : there exists C > 0 such that |� f, g �| ≤ C � g, − L 0 g � 1 / 2 ∀ g ∈ D ( L 0 ) Above �· , ·� is the scalar product in L 2 ( Q 0 ). Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
∂ λ =0 Q λ ( f ) Theorem Suppose E (e pZ 0 ) < ∞ for some p > 2 . Then, for any f ∈ H − 1 ∩ L 2 ( Q 0 ) , ∂ λ =0 Q λ ( f ) exists. Moreover: � �� � Q 0 k ∈ Z p 0 ,x k ( x k − ϕ ) h ∂ λ =0 Q λ ( f ) = − Cov( N f , N ϕ ) Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
Representation of ∂ λ =0 Q λ ( f ) by forms • Homogenization theory • M measure on Ω × Z � � � M ( u ) = Q 0 p 0 ,x k u ( ξ, k ) , u ( ξ, k ) Borel, bounded k • L 2 ( M ): square integrable forms • Potential form: g ∈ L 2 ( Q 0 ) ∇ g ( ξ, k ) := g ( τ k ξ ) − g ( ξ ) , • Given ε > 0 let g ε ∈ L 2 ( Q 0 ) solve ( ε − L 0 ) g ε = f • Kipnis–Varadhan [CMP, 1986]: ∇ g ε → h in L 2 ( M ) Alessandra Faggionato Ballisticity and Einstein relation in 1d Mott variable
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