User:
System Status
Description
To describe a physical system we must describe the state in which it is. This is to define the different parameters that describe the current situation and that can be used to predict how it can evolve over time.
In a classic system this is the positions and moment of all existing particles . These parameters represent the initial state and with the equations of motion it should be possible to predict any future state in which the system may be found.
In this description there are two problems:
The problem is that we don't know the initial state of the particles.
The number of equations to solve is too large.
ID:(518, 0)
Working with Multiple Copies of the System Status
Description
One of the problems of modeling a system of many particles is that we do not know its initial states, so we cannot predict its future evolution. One way to solve this problem is to assume that the
system can be in any of the possible initial states.
So instead of studying the evolution of a state,
we study the evolution of a set of states in which each one is initially in one of the possible states.
This set of states is called the statistical assembly.
The result for this reason will not be the final state of the system, but rather the set of final states of the states contained in the assembly.
The way to describe this result will be
indicating the probability that the system will end up in a particular state type.
ID:(519, 0)
Basic Postulate
Description
The basic postulate of statistical mechanics is that for
an isolated system in equilibrium exists the same probability of finding it in any of its states that it can access.
In this way, it is not necessary to know the state of a particular system and what is calculated is always valid for any system under the same conditions.
ID:(520, 0)
Statistical Ensemble
Description
The set of possible states is called an statistical assembly. The assembly is characterized by the conditions that define that the states are possible.
These conditions are generally macroscopic parameters such as energy, temperature, pressure, volume, etc. while the states themselves are defined by microscopic properties as the parameters of the phase space (moments and positions).
ID:(526, 0)
Probability calculation
Description
Since probability is defined as the ratio of favorable cases to possible cases, we can now define the probability of a specific type of state occurring. To do this, we need to narrow down the states to those associated with a characteristic $y_k$. With this in mind,
the probability of $y_k$ occurring will be equal to the proportion of the number of states that have the value $y_k$ relative to all possible states.
ID:(521, 0)
States to be accessed in the System
Description
When we talk about possible states we are indicating that there are parameters that define whether a state is physically possible or not.
Examples of this type of condition are the physical space that the particles can access and / or the total energy that the system has.
In an approximation that the particles do not interact, the first example applies to each particle independent of the others.
In the case of the energy of the system, the possible states are all those that present energy distributions among the particles such that the total energy defined is obtained in the sum of the energy.
If in the case of volume it is assumed that the particles themselves are impenetrable, a situation similar to that of energy occurs: only states are possible in which the particles are within the volume and also do not overlap.
Therefore, statistical assemblies are established based on the set of
all possible states that satisfy the constraints defined for the statistical assembly.
Once the assembly is established, it is determined
$\Omega(E,N)$ the number of states that meet the constraints that define the assembly (example total energy $E$ and number of particles $N$)
ID:(525, 0)
Probability of finding the system in a state
Equation
Una vez se ha determinado el ensamble estadístico se puede estimar la probabilidad de la ocurrencia de una situación en particular calculando el
numero $\Omega(E,N,y_k)$ de estados dentro del ensamble estadístico que cumplen adicionalmente la restricción $y_k$
con lo que la probabilidad de que ocurra
| $ P(E,N,y_k) =\displaystyle\frac{ \Omega(E,N,y_k) }{ \Omega(E,N) }$ |
ID:(11503, 0)
Expected value
Equation
Si la probabilidad es de encontrar el ensamble estadístico en el estado
| $ P(E,N,y_k) =\displaystyle\frac{ \Omega(E,N,y_k) }{ \Omega(E,N) }$ |
Este se calcula ponderando los valores con la probabilidad de un sistema tenga dicho valor, o sea con numero de estados con energía y partículas $-$, numero de estados con energía, partículas y parámetro $-$ and probabilidad de encontrar el sistema con energía, partículas y parámetro $-$ se tiene que:
| $\bar{y}=\displaystyle\frac{\sum_k\Omega(E,N,y_k)y_k}{\Omega(E,N)}$ |
ID:(3422, 0)
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Video
Video: System Ensamble