of a volume-preserving map is one for which because describes all the pieces-parts that came from. One is now interested in studying the time evolution of the system. If a set eventually comes to fill all of over a long period of time (that is, if approaches all of for large ), the system is said to be ergodic. If every set behaves in this way, the system is a conservative system, placed in contrast to a dissipative system, where some subsets wander away, never to be returned to. An example would be water running downhill: once it's run down, it will never come back up again. The lake that forms at the bottom of this river can, however, become well-mixed. The ergodic decomposition theorem states that every ergodic system can be split into two parts: the conservative part, and the dissipative part. Mixing is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets , and not just between some set and . That is, given any two sets , a system is said to be (topologically) mixing if there is an integer such that, for all and , one has that . Here, denotes set intersection and is the empty set. Other notions of mixing include strong and weak mixing, which describe the notion that the mixed substances intermingle everywhere, in equal proportion. This can be non-trivial, as practical experience of trying to mix sticky, gooey substances shows. Processes The above discussion appeals to a physical sense of a volume. The volume does not have to literally be some portion of 3D space; it can be some abstract volume. This is generally the case in statistical systems, where the volume (the measure) is given by the probability. The total volume corresponds to probability one. This correspondence works because the axioms of probability theory are identical to those of measure theory; these are the Kolmogorov axioms. The idea of a volume can be very abstract. Consider, for example, the set of all possible coin-flips: the set of infinite sequences of heads and tails. Assigning the volume of 1 to this space, it is clear that half of all such sequences start with heads, and half start with tails. One can slice up this volume in other ways: one can say "I don't care about the first coin-flips; but I want the 'th