We like to describe and understand the way things change. The mathematical approach of evolution processes usually consists in a description of the state of a given system at a given time together with a rule telling us how this state changes with time. The formulation of these rules is often quite complicated.
To give an example (you do not need to undestand the details!), we can
describe the state of fluid of constant density 
in terms of a
the velocity 
of the fluid at a position 
and a
time t, and the pressure 
.
Since we will not use these equations in a detailed way, the intuitive
meaning of density, velocity and pressure will be sufficient.
The evolution of the system can in
principle be determined by solving a partial differential equation (PDE)
known as the Navier-Stockes equation
The symbols 
and 
are called
partial derivatives. They tell us how the quantities on which
they are applied change in space and time.
The use of boldface characters such as
indicates a collection of variables, or more precisely a vector.
PDE are very useful to describe concrete systems but they are
usually very difficult to solve and we will discuss more ``idealized''
examples.
A simpler type of evolution is the classical mechanics description
of the motion of point-like
objects, where the state of one object can be described by a finite number
of variables: the
instantaneous position
and velocity 
. The time evolution of such a system
is described by ordinary differential equations (ODE). In the case
of a single particle the ODE can be written
where f is related to forces acting on the object
and 
is the time derivative, it tells us how
the quantities on which it is applied change in time. ODE
are discussed in the section 3 of Part 1 of the course.
Using a computer, one can solve ODE numerically. It is
usually not too difficult for a sufficiently short amount of time.
In the two above examples of evolution, time flows continuously.
On the other hand, one can also observe the state of the
system at discrete time and labels the successive observations
with 
. If as in the ODE case, it is possible
to specify the state of the system with a finite number of
variables, one might try to find a rule of evolution where the
state of the system at the n-th observation (denoted 
)
is all we need to know to predict the state of the system
at the n+1-th observation (denoted 
). Such rule
of evolution can be written
That such a law exists is not a totally absurd assumption. Indeed,
some numerical
solutions of ODE can be written in the form of Eq.(4).
The rule of evolution of Eq.(4) is clearly deterministic.
If we know the function 
and then is
uniquely fixed.
An important question which we want to address is the following. Given the fact that we can only know the state of the system at initial time with a limited precision, can we predict reliably the future of the system after an arbitrarily long time? And if not, is it nevertheless possible to say something useful about the evolution? Similar questions can be addressed for ODE or PDE.
Instead of starting with ODE or PDE, we shall first learn some basic concepts with simple examples which do not require calculus. In these examples, the state of a system is characterized by one or two numbers and a simple deterministic rule fixes its evolution. In the fifth example discussed, a probabilistic approach will be necessary. In other words, the best question we can ask is: what is the chance that the system is in a given state? The concepts developed with these examples will reappear naturally in the the description of systems with continuous evolution (based on differential equations), in particular Newtonian mechanics.
