Figure 1.
Discretization of the path integral.
The initial and final
variable are not integrated over.
Figure 2.
The probability distribution for the subcritical case () from the propagator (a) and from 2000 numerical experiment (b).
The solid line corresponds to T = 1, the dashed line to T = 2 and the dot-dashed line to T = 8.
Figure 3.
The time evolution of 10 members for the critical case(a) and the subcritical case
(b).
Figure 4.
The probability distribution for the subcritical case () from the propagator (a) and from 2000 numerical experiment (b).
The solid line correspond to T = 1, the dashed line to T = 2 and the dot-dashed line to T = 8.
Figure 5.
The propagators of the system: (a) the propagator for the variables (b) the propagator for the variables
.
A corresponding propagator can be obtained exchanging 2 and 4.
Figure 6.
(a) for the quadratic term , (b) for the the quartic term
.
Figure 7.
The graphical representation for the expressions (59) and (60).
Figure 8.
The terms of the perturbation expansion for the 2-point correlation, the variance.
The full contribution can be obtained by using symmetry over all the vertices and adding the graphs obtained exchanging 2 with 4: (a) disconnected graph, corresponding to (61), (b) graph with integrated over the internal vertex
, corresponding to (62), (c) graph with
into an external point corresponding to (63).
Figure 9.
The evolution of the equal time variance for an ensemble of 2000 simulations for the test system.
The averaged variance computed after equilibration and its standard deviation is shown to the right of the figure. The solid line represents the linear system, the dashed line is the non-linear system with and the dotted line is the non-linear system with
.