### Archive

Archive for January 26th, 2014

## A Remark based on False Claim 1

However, we can tell if time is passing so long as each Pasquali patch description is different at each (equal or unequal) interval.  If a single Pasquali patch were to describe the system at EACH distance interval, there is no way to know if time is moving at all.  Take for example the False Claim 1 of the previous post.  We had the collection $\mathbb{Q} = \{Q^1, Q^2, \ldots, Q^k, \ldots\}_{k \in \mathbb{Z}^+}$ with $Q = q(x,y)$ being an explicit function of $y$ and converging to $Q^\infty = 1$.  We could track the time-distance interval via the Pasquali patch power, so that $1$ was the first distance interval from start (we take it as given that time is not being bent, so that a fixed distance implies the passage of 1 unit of time), $2$ was the second distance interval (2 units of time), and so on, and we could tell if time were bent if each power were descriptive of different distance intervals.  Furthermore, since each $Q \in \mathbb{Q}$ is different, this implies each position transition probability is different and the system is in movement.
This is definitely not the case with the collection $\mathbb{P} = \{P^1 = 1, P^2 = 1, \ldots, P^j, \ldots \}_{j \in \mathbb{Z}^+}$ which also converges to $P^\infty = 1$.  Since at each distance interval the movement probability is the same (uniform), one cannot be convinced that each power represents a distance interval equal to equally spaced time intervals or different-spaced time intervals.  Where we could with $\mathbb{Q}$ ascertain that time was moving, we cannot with $\mathbb{P}$.  The statistical description cannot tell if the system is frozen.
When a system has reached the steady state (which, is the highest entropy state!), there is no way to tell if time flows, as the statistical description is and forever will be unchanging.  Recall that the steady state for a Pasquali patch is always a function of $x$ alone, say $p(x)$.  Furthermore recall that any power of $p(x)$ is always $p(x)$ itself (see Compendium).  We reach an impasse: is time flowing normally, faster than what is conventional, slower? At such a point it is impossible to say, at least from the statistical point of view.  We would have to track particles individually in order to ascertain if they have deviated their path at all (it could be the case that they shifted to all positions with equal probability or in the shape of $p(x)$, e.g., but we cannot be sure of either situation).