Bose-Einstein Distribution Plotter

Bose-Einstein Distribution Plotter visualizes boson occupancy vs. energy for given temperature and chemical potential, aiding quantum and statistical physics studies.

Temperature (K):

Chemical Potential (μ, eV):

Energy Range (ΔE, eV):

Formulas Used in Bose-Einstein Distribution Plotter

The plotter uses the following formula for the Bose-Einstein distribution:

Bose-Einstein Distribution:

\\[ f(E) = \frac{1}{e^{(E – \mu)/(k_B T)} – 1} \\]

Where:

\\( f(E) \\): Average number of bosons in a state (dimensionless, \\( \geq 0 \\))
\\( E \\): Energy (eV)
\\( \mu \\): Chemical potential (eV)
\\( k_B \\): Boltzmann constant (\\( 8.617333262 \times 10^{-5} \, \text{eV/K} \\))
\\( T \\): Temperature (K)

Example Calculations

Example 1: Photons at Room Temperature

Input: Temperature = 300 K, Chemical Potential = 0 eV, Energy Range = 0.5 eV

\\[ k_B T = 8.617333262 \times 10^{-5} \cdot 300 \approx 0.02585 \, \text{eV} \\] \\[ f(E = 0.1) = \frac{1}{e^{(0.1 – 0)/(0.02585)} – 1} \approx \frac{1}{e^{3.868} – 1} \approx 0.021 \\]

Result: Occupancy at E = 0.1 eV: 0.021, Plot Range: [0, 0.5] eV

Example 2: Low Temperature Bosons

Input: Temperature = 10 K, Chemical Potential = -0.01 eV, Energy Range = 0.5 eV

\\[ k_B T = 8.617333262 \times 10^{-5} \cdot 10 \approx 0.0008617 \, \text{eV} \\] \\[ f(E = 0) = \frac{1}{e^{(0 – (-0.01))/(0.0008617)} – 1} \approx \frac{1}{e^{11.604} – 1} \approx 8.63 \times 10^{-6} \\]

Result: Occupancy at E = 0 eV: 8.63e-6, Plot Range: [0, 0.5] eV

Example 3: Wide Energy Range

Input: Temperature = 300 K, Chemical Potential = -0.1 eV, Energy Range = 1 eV

\\[ k_B T = 8.617333262 \times 10^{-5} \cdot 300 \approx 0.02585 \, \text{eV} \\] \\[ f(E = 0) = \frac{1}{e^{(0 – (-0.1))/(0.02585)} – 1} \approx \frac{1}{e^{3.868} – 1} \approx 0.021 \\]

Result: Occupancy at E = 0 eV: 0.021, Plot Range: [0, 1] eV

Formulas Used in Bose-Einstein Distribution Plotter

Example Calculations

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