7. Proposed Methods 65
8.1. Reaction Diusion System
8. Microscopic Models
In this chapter I apply the Schur DMRG method, presented in Section 7.1, to some simple one-dimensional models that are described by a microscopic dynamics.
Technically these are modelled via the master equation as described in Chapter 4.
The focus here is set on a systematic treatment of in principle nonlinear models.
By the conversion to a high-dimensional linear model one can rely on the known results for such systems and need not to treat the nonlinearity approximatively.
All models in this section are stochastic already by construction.
For the study of stochastic non-equilibrium systems DMRG has been previously employed [28, 38, 27]. There the focus was on the steady state. The calculation of the long living transient states have proved to be a numerically demanding task. In the following the sorted Schur vectors will be calculated, which give also a description for the long time transient behaviour, but are more appropriate for a description as explained in Chapter 5.
8. Microscopic Models
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Steady state
Position
<n>
N=6 m=4 sweeps:0 N=10 m=6 sweeps:2 N=12 m=6 sweeps:2 N=14 m=8 sweeps:2 N=16 m=14 sweeps:1 N=18 m=16 sweeps:1
Figure 8.1.: Rescaled average occupation number for various lattice sizes in the steady state of the reaction diusion system. The state was nor-malised according to Eq.(4.2), i.e. P
Ψi = 1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.05
−0.045
−0.04
−0.035
−0.03
−0.025
−0.02
−0.015
−0.01
−0.005 0
Steady state
Position
<n
in
i+1>
N=6 m=4 sweeps:0 N=10 m=6 sweeps:2 N=12 m=6 sweeps:2 N=14 m=8 sweeps:2 N=16 m=14 sweeps:1 N=18 m=16 sweeps:1
Figure 8.2.: Rescaled nearest neighbour density correlation for various lattice sizes in the steady state of the reaction diusion system. The state was normalised according to Eq.(4.2), i.e. P
Ψi = 1.
8.1. Reaction Diusion System
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
1.Transient state
Position
<n>
N=6 m=4 sweeps:0 N=10 m=6 sweeps:2 N=12 m=6 sweeps:2 N=14 m=8 sweeps:2 N=16 m=14 sweeps:1 N=18 m=16 sweeps:1
Figure 8.3.: Rescaled average occupation number for various lattice sizes in the rst transient state of the reaction diusion system. The state was normalised according to Eq.(4.2), i.e. P
Ψi = 1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.2
−0.18
−0.16
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02 0
1.Transient state
Position
<n
in
i+1>
N=6 m=4 sweeps:0 N=10 m=6 sweeps:2 N=12 m=6 sweeps:2 N=14 m=8 sweeps:2 N=16 m=14 sweeps:1 N=18 m=16 sweeps:1
Figure 8.4.: Rescaled nearest neighbour density correlation for various lattice sizes in the rst transient state of the reaction diusion system. The state was normalised according to Eq.(4.2), i.e. P
Ψi = 1.
8. Microscopic Models
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2.Transient state
Position
<n>
N=6 m=4 sweeps:0 N=10 m=6 sweeps:2 N=12 m=6 sweeps:2 N=14 m=8 sweeps:2 N=16 m=14 sweeps:1 N=18 m=16 sweeps:1
Figure 8.5.: Rescaled average occupation number for various lattice sizes in the second transient state of the reaction diusion system. The state was normalised according to Eq.(4.2), i.e. P
Ψi = 1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1
2.Transient state
Position
<n
in
i+1>
N=6 m=4 sweeps:0 N=10 m=6 sweeps:2 N=12 m=6 sweeps:2 N=14 m=8 sweeps:2 N=16 m=14 sweeps:1 N=18 m=16 sweeps:1
Figure 8.6.: Rescaled nearest neighbour density correlation for various lattice sizes in the second transient state of the reaction diusion system. The state was normalised according to Eq.(4.2), i.e. P
Ψi = 1.
8.1. Reaction Diusion System
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
Position
<n>
Steady state 1.Transient 2.Transient
Figure 8.7.: Rescaled average occupation number for the reaction diusion system and the6-site lattice in the rst eigenstates. Solid lines: Results from the Schur-DMRG method, dashed lines: POD modes. The curves for the steady state lie on top of each other. Note that the transient states can be interpreted as corrections to the steady state. Therefore negative values are permitted.
to a PDE. By mapping the model onto a quantum chain analytical results were obtained in [1].
Without the source term the system tends to an empty lattice for large times for all initial conditions [27]. It is known that for low spatial dimensions the diusion is ecient for mixing [1].
8.1.1. Numerical Results
The rst three Schur vectors are calculated for the parameters Ks = 1, Ka = 1 and Kd = 1. To visualise the results, the average density prole and the nearest neighbour density correlation are evaluated.
Normalisation of the Results
For the reaction diusion model the normalisation condition X
i
Ψi = 1, (8.4)
is adopted for all statesΨ. This choice accounts for the probabilistic interpretation of the state vectorΨ. The transient states can be considered as corrections to the steady state so that these states can have negative components. In principle then
8. Microscopic Models also the normP
iΨi can be zero. However, this feature is not present in the calcu-lations on the reaction diusion model so one can use the normalisation Eq.(8.4).
Nevertheless the situation P
iΨi = 0 will be encountered in later sections.
Average Density Prole
The density proles for lattice sizes from 6 to 18 for the steady state are shown in Fig 8.1. The corresponding phase spaces were 64 to 262144-dimensional. The prole shows fast decrease near the source site, due to the annihilation and a long tail. The proles also show signs for numerical inaccuracies. For the system of size 6 (with m = 4) the Schur-DMRG procedure is equivalent to a direct real Schur decomposition. The same data for the rst transient state is presented in Fig. 8.3. Note that while the Schur vectors are orthogonal, the density proles for dierent states are not. Comparing Fig. 8.1 and Fig. 8.3 one sees that the long time corrections are most important for the region which has a low average occupation. The second transient state gives a very similar correction to the density, although all Schur vectors are mutually orthogonal. For larger system sizes also the correction to the steady state are smaller. The correct eigenstates are derived from the Schur vectors by diagonalising the eective master operator M as
B†MBV =VM, (8.5)
where B contains the Schur vectors and V is a matrix with normalised columns.
The entries for one column of V are the expansion coecients for the eigenstates in the Schur vectors. The resulting density proles are shown in Fig. 8.7.
To evaluate the results a direct simulation of the model has been performed.
In order to calculate an average density, an ensemble of 2000 random (uniformly distributed) initialised states were evolved under the stochastic time evolution for 106 time steps. For the resulting time dependent density prole a proper orthogonal decomposition was performed. The results are shown in Fig. 8.7 as dashed lines. The agreement for the steady state is excellent. For the transient states this is clearly not the case. This is due to the fact, that the density proles of the Schur vectors as well as of the eigenvectors are not orthogonal. On the other hand, the POD-modes are by construction orthonormal. The density prole further does not contain all information on the stochastic process. Therefore the failure of this comparison does not question the Schur DMRG results.
Nearest Neighbour Density Correlation
The nearest neighbour density correlation ci is dened by
ci =< nini+1 >−< ni >< ni+1 > , i= 1 :N −1. (8.6) This function has been evaluated for lattice sizes from6to18. For the steady state one observes a negative correlation for all positions, see Fig. 8.2. The absolute