Statistical Thermodynamics – Introduction

What is thermodynamics?

  • Describes the behavior of matter and the transformation between different forms of energy on a macroscopic scale.
  • Systems are described in terms of their bulk properties (e.g., pressure, volume, mass).
  • It is based on three empirical laws of experience (phenomenological).
  • A totally self-consistent description of the macroscopic properties of a system.

What is statistical thermodynamics?

  • Establishes a connection between the macroscopic properties of a system and its molecular properties.
  • Allows prediction of thermodynamic properties from structural and spectroscopic data.
  • Provides insight into the molecular origin of chemical properties, linking structure to function.

Boltzmann's entropy equation is the fundamental equation quantitatively reconciles the microscopic probabilistic world with the deterministic macroscopic laws of thermodynamics.:

\[ S = k \ln W \]

Here, $W$ represents the number of microstates, and $k_B$ is the Boltzmann constant.

 

Why Statistical Thermodynamics?

Statistical thermodynamics provides a robust framework for understanding the molecular basis of macroscopic thermodynamic properties. It links the average behavior of vast numbers of particles to observable phenomena, bridging the gap between microscopic and macroscopic worlds.

  • Microscopic world: Atoms, molecules, photons, and electrons occupy a very large number of states (energy levels).
  • Macroscopic world: Behavior results from the average properties of microscopic entities, necessitating statistical analysis.

For example, the macroscopic variable A can be expressed as:

\[ A = \sum_{j} A(E_j) \cdot p(E_j) \]

where $A(E_j)$ is the property at a given energy level, and $p(E_j)$ is the probability of a given energy level.

 

Key Vocabulary in Statistical Thermodynamics

Microstate: A specific microscopic configuration representing the number of entities/particles in a given energy level (state).

Macrostate: Specifies the number of particles in each energy level, encompassing many arrangements of microstates.

Probability function: Illustrates the likelihood that each entity/particle will be in a particular state.

Familiar statistical Example 1: Coin Tosses

Consider a system with four coins. Each coin can be in one of two states: Heads (H) or Tails (T).

The possible macrostates and the number of microstates for each macrostate are:

Macrostate Number of Heads Number of Tails

Thermodynamic probability

$W_k$

Mathematical probability

$P_k = \frac{W_k}{\sum_k W}$
HHHH 4 0
\[1\]
\[\frac{1}{16}\]
HHHT 3 1
\[4\]
\[\frac{4}{16}\]
HHTT 2 2
\[6\]
\[\frac{6}{16}\]
HTTT 1 3
\[4\]
\[\frac{4}{16}\]
TTTT 0 4
\[1\]
\[\frac{1}{16}\]

This demonstrates how probabilities and microstates are connected in statistical thermodynamics.

 

The calculation of average occupation or the average number of an outcome in all macrostates can be demonstrated again with the heads and tails example. The equation for the average number of heads ($\bar{N}_H$) in this example is:

\[\bar{N}_H = \frac{\sum_{k=1}^{5} N_{H,k} W_k}{\sum_{k=1}^{5} W_k} = \sum_k N_{H,k} \cdot P_k\]

Here, $N_{H,k}$ is the number of heads in macrostate $k$, $W_k$ is the number of microstates corresponding to macrostate $k$, $P_k$ is the probability of macrostate $k$, defined as $P_k = \frac{W_k}{\sum W_k}$.

Substituting the values from the table of macrostates and microstates:

\[\bar{N}_H = \frac{(4 \cdot 1) + (3 \cdot 4) + (2 \cdot 6) + (1 \cdot 4) + (0 \cdot 1)}{16} = \frac{4 + 12 + 12 + 4 + 0}{16} = 2\]

  • The numerator sums the product of the number of heads ($N_H$) and the number of occurrences ($W_k$).
  • The denominator sums the total occurrences ($\sum W_k$).

Interpretation: The average number of heads across all macrostates is $2$, consistent with a binomial distribution of outcomes.

The diagram represents the distribution of macrostates ($(4,0)$, $(3,1)$, etc.) and their corresponding microstates for a system with 4 coins. The bar graph shows the probability distribution of #P_k#.

The number of microstates ($W_k$) represents the likelihood that each entity/particle will be in a particular macrostate. It is equivalent to the number of ways to arrange particles into specific configurations (e.g., heads and tails). This can be calculated using the following formula:

\[W_k = \binom{N}{N_H} = \frac{N!}{N_H! (N - N_H)!}\]

Where $N$ is the total number of particles (e.g., number of coin tosses), $N_H$ is the number of heads (or desired outcome), $N - N_H$ is the number of tails (or other outcomes), $N!$ is the factorial of $N$, defined as $N! = N \cdot (N - 1) \cdot (N - 2) \cdot \dots \cdot 1$. For special cases, $0! = 1$.

Examples:

1. For $N = 4$ and $N_H = 3$:

\[W_k = \binom{4}{3} = \frac{4!}{3! \cdot 1!} = \frac{4 \cdot 3 \cdot 2 \cdot 1}{(3 \cdot 2 \cdot 1) \cdot 1} = 4\]

This means there are 4 ways to have 3 heads in 4 tosses.

2. For $N = 10$ and $N_H = 5$,

\[W_k = \binom{10}{5} = \frac{10!}{5! \cdot 5!} = 252\]

As the number of particles increases ($N \sim 10^3$, $10^6$, etc.), the calculation of factorials for very large numbers becomes computationally intensive.

Stirling's Approximation: An accurate method for approximating factorials of very large numbers ($a!$), commonly encountered in statistical thermodynamics.

Background:

  • Factorial ($a!$) grows very rapidly for large numbers (e.g., $a \sim 10^2$). Direct computation becomes impractical.
  • The logarithm of a factorial can be approximated using an integral: $\ln a! = \sum_{r=1}^a \ln r \approx \int_1^a \ln x \, dx$

Derivation Using Integration by Parts:

  • Set $u = \ln x$ and $dv = dx$, so $du = \frac{1}{x} dx$ and $v = x$.
  • The integral becomes: $\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} \, dx = x \ln x - x + C$.

Applying this to the factorial: $\ln a! \approx a \ln a - a + 1$

The result can be converted to to Base-10 Logarithms using the relationship between natural logarithms and base-10 logarithms: $\log_{10} a! = \frac{\ln a!}{\ln 10}$. Thus, the final form of Stirling's approximation becomes:

\[a! \approx 10^{0.434 \cdot (a \ln a - a + 1)}\]
.

The graph illustrates the approximation of $\ln a!$ as the area under the curve $y = \ln a$. The shaded regions represent the integration steps.

 

Number of Microstates in Multiple State Systems

Definition: The number of microstates ($W$) represents the arrangement of distinguishable objects in groups corresponding to different macrostates.

General Formula:

\[ W = \frac{N!}{N_1! \cdot N_2! \cdot N_3! \cdots}. \]

Explanation: This formula accounts for the total number of ways to arrange $N$ objects, divided by the permutations of objects within each group ($N_1$, $N_2$, $N_3$, etc.).

Example Calculation for $n = 3$:

  • Step 1: Divide the objects into three groups: $ W = \frac{N!}{N_1! (N-N_1)!}, \quad W = \frac{(N - N_1)!}{N_2!(N-N_1-N_2)!} = \frac{(N - N_1)!}{N_2!N_3!}, \quad W = \frac{(N - N_1 - N_2)!}{N_3!(N-N_1-N_2-N_3)!} = \frac{N_3!}{N_3!}$.
  • Step 2: Combine these terms: $ W_{\text{total}} = \frac{N!}{N_1! \cdot N_2! \cdot N_3!}$.

Interpretation: The total number of microstates for $n = 3$ distinguishable objects is given by: $ W_{\text{total}} = \frac{N!}{N_1! \cdot N_2! \cdot N_3!}$.

This calculation is critical in statistical thermodynamics for understanding the distribution of particles among energy levels or states. It provides the basis for calculating entropy and probability distributions in macrostates.

 

Averaged behavior in Statistical Thermodynamics

Statistical thermodynamics focuses on predicting the average or most likely outcome for a given phenomenon. This approach is critical because individual behaviors at the microscopic level (e.g., atoms and molecules) average out to produce observable macroscopic properties.

  • System Tendency: A system naturally tends towards occupying the most probable state, which corresponds to:
    • Highest disorder: Systems prefer disordered states rather than ordered ones due to the statistical dominance of disordered configurations.
    • Highest entropy: Entropy represents the level of disorder or randomness in a system. Systems evolve spontaneously to states of maximum entropy.
    • Thermodynamic equilibrium: This is the state where the macroscopic properties of the system no longer change with time, reflecting maximum entropy and stability.

The diagrams depict key processes:

  • The left-hand sketch shows particles moving from an organized to a disorganized distribution, aligning with the tendency for systems to favor high-disorder states.
  • The right-hand diagram illustrates thermal equilibration between a hot and cold region, representing the natural flow of energy until a uniform temperature is reached.

These examples highlight that macroscopic equilibrium is driven by microscopic-level interactions and randomness.

To describe the system's average behavior, we use the statistical concept of averaging over all possible states. This is formalized by the equation:

\[ \langle A \rangle = \sum_j A(E_j) \cdot p(E_j) \]

Here $\langle A \rangle $ represents the average value of a property $A$ over all possible energy levels, $A(E_j)$ the value of property $A$ at a specific energy level $E_j$, $p(E_j)$ is the probability of the system being in energy level $E_j$

In a large system, molecules occupy various energy levels based on their interactions and environmental conditions. Since these energy levels are probabilistic, we must average their contributions to calculate any macroscopic property:

  • For example, the temperature of a gas in a room is the average kinetic energy of all molecules, even though individual molecules have varying speeds.
  • Similarly, pressure and volume result from averaging molecular collisions and positions over time.

Example:

Consider a gas system where particles occupy multiple energy levels. To calculate a property such as the average kinetic energy ($\langle KE \rangle $), we compute:

\[ \langle KE \rangle = \sum_j KE(E_j) \cdot p(E_j) \]

Using statistical distributions $p(E_j)$ can be derived based on the temperature and energy levels, enabling precise macroscopic predictions.