Elementary Statistical Mechanics

 Good evening to you all

Today in my blog I will talk about yesterday’s topic which is elementary statistical mechanics.

Elementary Statistical Mechanics

Elementary statistical mechanics is a branch of physics that deals with the study of the behavior of macroscopic systems by analyzing the properties of their microscopic constituents. American physicist Josiah Willard Gibbs proposed the mathematical structure of statistical mechanics in the late 19th century.

There are two main approaches to statistical mechanics:

1)      Classical Mechanics

2)      Quantum Mechanics

Classical Mechanics

Classical mechanics is based on the classical laws of physics, which deal with the study of macroscopic bodies.

Quantum Mechanics

 Quantum Mechanics is the study of microscopic bodies such as subatomic particles, atoms, and other small bodies.

In Elementary statistical mechanics, a system can be described in terms of its micro-state and macro-state.

Micro-state

The arrangement of particles in different ways is known as a micro-state. It is a specific configuration of the system's particles, including their positions, velocities, and energies. Each microstate is unique and corresponds to a particular arrangement of the particles in the system. Classical thermodynamics describes that thermodynamic systems are macroscopic systems containing macroscopic properties. However, all these thermodynamic systems are made up of atoms; therefore, it is very important to understand the microstate of the system as well, which specifies the quantum state of all the atoms in the system.

For example, if four distinguishable particles are distributed in two compartments, then, the no. of possible microstates (16)                                

=2^4=16

If n particles are to be distributed in 2 compartments. The no. of microstates is

=2^n

Macro-state

The number of arrangement of particles is known as a macro-state. It is a collection of microstates that share certain macroscopic properties, such as the total energy, volume, and number of particles in the system. In other words, a macro-state is a set of micro-states that are indistinguishable from a macroscopic perspective.

For example, if 4 particles are distributed in 2 components, then the possible macrostates i.e., 4+1=5.

If n particles are to be distributed in 2 components. Then the no. of macrostates is 

=n+1                                  

 Differences between micro-state and macro-state:

The relationship between micro-state and macro-state is important in statistical mechanics because it allows us to relate the microscopic behavior of the particles to the macroscopic properties of the system. By understanding the distribution of the micro-state that corresponds to a particular macro-state, we can make predictions about the behavior of the system as a whole.

Thermodynamic probability:

In thermodynamics, the concept of probability is used to describe the likelihood of a system being in a certain state. The thermodynamic probability of a system in a given state is proportional to the number of microstates that correspond to that macrostate. OR The number of microstates corresponding to any macrostates is known as thermodynamic probability. The thermodynamic probability of the system being in a particular macrostate is given by the following equation:

Where P is the thermodynamic probability, Ω is the number of microstates that correspond to the macrostate, and N is the total number of particles in the system.

For distribution of 4 particles in 2 identical compartments

Distribution of four indistinguishable particles in two components.

For indistinguishable particles, W=1

Four particles can be distributed among the two components.

Microstates		Macrostates	Frequency Or Thermodynamic Probability W	Probability
Component 1	Component 2
abcd		(4,0)	1	1/16
bcd acd abc	a b c a	(3,1)	4	4/16
ab ac ad bc bd cd	cd bd bc ad ac ab	(2,2)	6	6/16
a b c d	bcd acd abd abc	(1,3)	4	4/16
	abc	(0,4)	1	1/16

 Thank You 😍