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8 Quantum Ratchets
h2=zeros(N-1,NR);
eps3=zeros(N-1,NR);
h3=zeros(N-1,NR);
gw=zeros(N-1,NR);
a1 = abs(randn(1));
b1 = abs(randn(1));
a2 = abs(randn(1));
b2 = abs(randn(1));
a3 = abs(randn(1));
b3 = abs(randn(1));
E1 = exp(-(dt/(t1*tR)));
E2 = exp(-(dt/(t2*tR)));
E3 = exp(-(dt/(t3*tR)));
xij=x(1,1);
%U(i,j)=-(Uo/(2*pi))*(sin(2*pi*xij/L) - 0.25*sin(4*pi*xij/L));
%Potential tilded to the left.
%Dd1U(1,1)=(Uo/L)*(cos(2.*pi*xij) - 0.5*cos(4.*pi*xij))*(L/DU);
Dd2U(1,1)=-Uo*(2.*pi/L 2 )*(-sin(2.*pi*xij) + sin(4.*pi*xij))*(L 2 /DU);
%Dd3U(1,1)=Uo*((2.*pi 2 )/L 3 )*(-cos(2.*pi*xij) + 2.*cos(4.*pi*xij))*(L 3 /DU);
DDeff(1,1)=1./(1-Dlambda*Dbeta*Dd2U(1,1));
%DDeff(1,1)=D;
eps1(1,1) = sqrt((-2.*(DDeff(1,1)*D1)/(t1*tR))*(1-E1 2 )*log(a1))*cos(2*pi*b1);
eps2(1,1) = sqrt((-2.*(DDeff(1,1)*D2)/(t2*tR))*(1-E2 2 )*log(a2))*cos(2*pi*b2);
eps3(1,1) = sqrt((-2.*(DDeff(1,1)*D3)/(t3*tR))*(1-E3 2 )*log(a3))*cos(2*pi*b3);
eps(1,1)=eps1(1,1)+eps2(1,1)+eps3(1,1);
for j=1:NR
for i=1:N-1
xij=x(i,j);
Dd1U(i,j)=-(Uo/L)*(cos(2.*pi*xij) - 0.5*cos(4.*pi*xij))*(L/DU);
Dd2U(i,j)=-Uo*(2.*pi/L 2 )*(-sin(2.*pi*xij) + sin(4.*pi*xij))*(L 2 /DU);
Dd3U(i,j)=-((Uo*(2.*pi) 2 )/L 3 )*(-cos(2.*pi*xij) + 2.*cos(4.*pi*xij))*(L 3 /DU);
DDeff(i,j)=1./(1-Dlambda*Dbeta*Dd2U(i,j));
%========================================================
% QUANTUM LANGEVIN EQUATION
%========================================================
x(i+1,j)=xij - Dd1U(i,j)*dt - 0.5*Dlambda*Dd3U(i,j)*dt +eta(i,j)*FDN*dt. . .
+ Fext*dt + eps(i,j)*dt;
v(i+1,j)=(x(i+1,j)-xij)/dt;
%========================================================
% NOISE %================================================