I think that this observation could explain why we see a reversible 
all the irreversibility has already happened. If we think of a dynamics 
discrete time then we have a collection of points with directed arcs
between them. As a graph, this has the structure of several cycles with
trees connected to some of the points. The trees correspond to the 
part of the dynamics, the cycles to the reversible part.

If the largest tree is of height h, then after h time steps, the system
must be in a state on one of the cycles. Thus the dynamics is 

Of course this argument requires a finite state system, which is usually
assumed in such discussions. An uncountably infinite counterexample to 
this idea
is an infinite tree, with every node branching to two predecessors. At 
state and every time step there is an irreversible transition.

A countable counterexample can be assembled by grafting a copy of the 
natural numbers
onto the integers with the system state transition taking n to n-1. 
Then 0
has two predecessors. Because there is no bound on the time taken for a 
pair of
distinct states (the same positive integer on the two branches) to be 
mapped together,
the reversibility does not dissipate.

I thought I had a copy of the paper here, but I cannot locate it. If 
serves me right, it was one of a series of papers that Toffoli wrote in 
last half of the 90s dealing with computation and physics. Most of them 
good reading anyway, so have a dive into:


On Apr 11, 2006, at 1:19 AM, Wei Dai wrote:

> Ti Bo wrote:
>> On reversibility, there is the observation (I think acredittable to 
>> Tom
>> Toffoli)
>> that most/all irreversible systems have a reversible subsystem and the
>> dynamics arrive in that
>> subsystem after some (finite) time. Thus any system that we observe a
>> while
>> after it has started will, with high likelihood, be reversible. In 
>> some
>> sense the
>> irreversibility dissipates and leaves a reversible core.
> That's an interesting observation, but are you suggesting that it can
> explain why our universe is reversible? If so, how? Do you have a 
> reference
> to a fuller explication of the idea?
> >
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