un:bose-einstein-statistics
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| un:bose-einstein-statistics [2025/11/30 07:33] – asad | un:bose-einstein-statistics [2025/11/30 07:37] (current) – asad | ||
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| - | Here the factor of 2 accounts for two photon polarization states. For massive bosons, this factor is replaced by the appropriate spin degeneracy. This function expresses a key difference between classical and quantum statistics: the denominator can become small, allowing | + | Here the factor of 2 accounts for two photon polarization states. For massive bosons this factor is replaced by the appropriate spin degeneracy |
| - | For photons, the variable \( \nu \) labels frequency, and this same distribution underlies the Planck spectrum of [[Blackbody radiation]]. For massive bosons, the energy is instead \( \epsilon = p^{2}/2m \), but the statistical form is identical: | + | For photons, the variable \( \nu \) labels frequency, and this same distribution underlies the Planck spectrum of [[Blackbody radiation]]. For massive bosons, the energy is instead \( \epsilon = p^{2}/(2m) \), but the statistical form is identical: |
| $$ | $$ | ||
| - | f = \frac{g}{h^{3}} \frac{1}{e^{(\epsilon-\mu)/ | + | f = \frac{g}{h^{3}} |
| $$ | $$ | ||
| - | with \( g \) the spin degeneracy. | + | The form of \( f \) therefore unifies the physics of blackbody photons and ideal Bose gases. |
| ===== Mean occupation of quantum states ===== | ===== Mean occupation of quantum states ===== | ||
| - | The phase-space distribution function determines the **mean number of bosons occupying each single-particle quantum state**. Integrating \(f\) over a cell associated with the state \(i\) leads directly | + | The phase-space distribution function determines the mean number of bosons occupying each single-particle quantum state. Integrating \( f \) over a phase-space |
| $$ | $$ | ||
| - | \bar{n}_i = \frac{1}{\exp\!\left(\frac{\epsilon_i - \mu}{kT}\right) | + | \bar{n}_i = \frac{1}{e^{(\epsilon_i - \mu)/(kT)} - 1}. |
| $$ | $$ | ||
| - | This quantity plays the same role for massive bosons that the photon distribution function plays for radiation. Summing over all states, | + | This quantity plays the same role for massive bosons that the photon distribution function plays for radiation. Summing over all states |
| $$ | $$ | ||
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| $$ | $$ | ||
| - | In the continuum limit with density of states \(g(\epsilon)\), | + | In the continuum limit with density of states \( g(\epsilon) \), |
| $$ | $$ | ||
| n = \frac{N}{V} | n = \frac{N}{V} | ||
| - | = \int_0^\infty \frac{g(\epsilon)}{\exp[(\epsilon-\mu)/ | + | = \int_0^\infty \frac{g(\epsilon)}{e^{(\epsilon-\mu)/ |
| $$ | $$ | ||
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| $$ | $$ | ||
| - | the Bose–Einstein distribution reduces to the Maxwell–Boltzmann form, | + | the Bose–Einstein distribution reduces to the Maxwell–Boltzmann form |
| $$ | $$ | ||
| - | \bar{n}_i \approx | + | \bar{n}_i \approx |
| $$ | $$ | ||
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| $$ | $$ | ||
| - | f \approx \frac{g}{h^{3}} e^{-(\epsilon-\mu)/ | + | f \approx \frac{g}{h^{3}} |
| $$ | $$ | ||
| - | Quantum statistical effects appear when the thermal de Broglie wavelength, | + | Quantum statistical effects appear when the thermal de Broglie wavelength |
| $$ | $$ | ||
| \lambda_{\mathrm{th}} | \lambda_{\mathrm{th}} | ||
| - | = \sqrt{\frac{2\pi\hbar^{2}}{m kT}}, | + | = \sqrt{\frac{2\pi\hbar^{2}}{m kT}} |
| $$ | $$ | ||
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| $$ | $$ | ||
| - | n\lambda_{\mathrm{th}}^{3} \gtrsim 1. | + | n \lambda_{\mathrm{th}}^{3} \gtrsim 1. |
| $$ | $$ | ||
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| ===== Bose–Einstein condensation ===== | ===== Bose–Einstein condensation ===== | ||
| - | For a uniform ideal Bose gas, the chemical potential satisfies \( \mu \le \epsilon_{0} \). As the gas is cooled at fixed density, \( \mu \rightarrow | + | For a uniform ideal Bose gas, the chemical potential satisfies \( \mu \le \epsilon_{0} \). As the gas is cooled at fixed density, \( \mu \to \epsilon_{0} \), and the excited states are no longer capable of holding all particles. The excess particles accumulate in the ground state, producing macroscopic occupation, |
| $$ | $$ | ||
| - | \bar{n}_{0} \rightarrow \text{large}. | + | \bar{n}_{0} \gg 1. |
| $$ | $$ | ||
| - | This is **Bose–Einstein condensation**, a striking manifestation of the Bose–Einstein distribution function. The condensation phenomenon is therefore the direct consequence of the form of \(f\) and the associated occupation numbers \(\bar{n}_i\): | + | This is Bose–Einstein condensation, |
| In contrast, for photons the chemical potential is fixed to zero, and instead of condensing, photons redistribute themselves to form the blackbody spectrum. | In contrast, for photons the chemical potential is fixed to zero, and instead of condensing, photons redistribute themselves to form the blackbody spectrum. | ||
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| ===== Bose–Einstein occupation curves ===== | ===== Bose–Einstein occupation curves ===== | ||
| - | To analyze how the Bose–Einstein distribution behaves, it is useful to define dimensionless variables, | + | To analyze how the Bose–Einstein distribution behaves, it is useful to define dimensionless variables |
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| $$ | $$ | ||
| - | Small \(a\) (i.e., large \( \mu \)) enhances the low-energy peak of the distribution. The special case \(a=0\) corresponds exactly to the photon distribution function | + | Small \( a \) (that is, large \( \mu \)) enhances the low-energy peak of the distribution. The special case \( a = 0 \) corresponds exactly to the photon distribution function |
| < | < | ||
| Line 134: | Line 134: | ||
| <div style=" | <div style=" | ||
| - | Parameter | + | Parameter a = -μ/(kT): |
| <input type=" | <input type=" | ||
| | | ||
| Line 140: | Line 140: | ||
| </ | </ | ||
| - | <script type=" | ||
| - | window.PlotlyConfig = {MathJaxConfig: | ||
| - | </ | ||
| <script src=" | <script src=" | ||
| <script type=" | <script type=" | ||
| //< | //< | ||
| + | |||
| + | // Linear x-grid for BE occupation | ||
| const Npts = 400; | const Npts = 400; | ||
| const xMin = 0.05; | const xMin = 0.05; | ||
| const xMax = 8.0; | const xMax = 8.0; | ||
| - | const x_vals = Array.from({length: | + | const x_vals = Array.from({length: |
| + | | ||
| + | ); | ||
| - | function nbar(x, | + | // Bose–Einstein occupation function |
| - | const e = Math.exp(x+a); | + | function nbar(x, a) { |
| - | return (e< | + | const e = Math.exp(x + a); |
| + | return (e <= 1) ? Infinity : 1 / (e - 1); | ||
| } | } | ||
| - | function makeBETrace(a){ | + | // Create a trace for a given " |
| + | function makeBETrace(a) { | ||
| return { | return { | ||
| - | x:x_vals, | + | x: x_vals, |
| - | y: | + | y: x_vals.map(x => nbar(x, a)), |
| - | mode:' | + | mode: ' |
| - | line: | + | line: {color: '# |
| + | name: 'a = ' + a.toFixed(1) | ||
| }; | }; | ||
| } | } | ||
| + | // Annotation with plain-text equation | ||
| const eq_be = { | const eq_be = { | ||
| - | x:0.97, y:0.97, xref:' | + | x: 0.97, y: 0.97, xref: ' |
| - | text:'$\\\\bar{n}(x;a)=1/(e^{x+a}-1)$', | + | text: 'n̄(x; a) = 1 / (exp(x + a) - 1)', |
| - | showarrow: | + | showarrow: false, |
| + | | ||
| + | | ||
| }; | }; | ||
| + | // Axis labels with no MathJax | ||
| const layout_be = { | const layout_be = { | ||
| - | margin: | + | margin: {l: 95, r: 20, t: 10, b: 60}, |
| - | xaxis: | + | xaxis: { |
| - | yaxis: | + | |
| - | annotations: | + | |
| + | | ||
| + | yaxis: { | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | annotations: | ||
| }; | }; | ||
| - | Plotly.newPlot(' | + | // Initial plot |
| + | Plotly.newPlot(' | ||
| - | const aslider=document.getElementById(' | + | // Slider control |
| - | const Aval=document.getElementById(' | + | const aslider = document.getElementById(' |
| + | const Aval = document.getElementById(' | ||
| - | aslider.oninput=function(){ | + | aslider.oninput = function() { |
| - | const a=parseFloat(this.value); | + | const a = parseFloat(this.value); |
| - | Aval.textContent=a.toFixed(1); | + | Aval.textContent = a.toFixed(1); |
| - | Plotly.react(' | + | Plotly.react(' |
| }; | }; | ||
| + | |||
| //]]> | //]]> | ||
| </ | </ | ||
| Line 198: | Line 216: | ||
| * Large low-energy occupation arises because the denominator may become small as \( \mu \to \epsilon_{0} \). | * Large low-energy occupation arises because the denominator may become small as \( \mu \to \epsilon_{0} \). | ||
| * Classical statistics emerges only when \( e^{(\epsilon-\mu)/ | * Classical statistics emerges only when \( e^{(\epsilon-\mu)/ | ||
| - | * Quantum degeneracy appears when \( n\lambda_{\mathrm{th}}^{3} \gtrsim 1 \), forcing the use of Bose–Einstein statistics. | + | * Quantum degeneracy appears when \( n \lambda_{\mathrm{th}}^{3} \gtrsim 1 \), forcing the use of Bose–Einstein statistics. |
| * Bose–Einstein condensation results directly from the inability of excited states to hold all particles as \( \mu \to \epsilon_0 \). | * Bose–Einstein condensation results directly from the inability of excited states to hold all particles as \( \mu \to \epsilon_0 \). | ||
| * For photons, the condition \( \mu = 0 \) transforms the Bose–Einstein distribution into the Planck spectrum of blackbody radiation. | * For photons, the condition \( \mu = 0 \) transforms the Bose–Einstein distribution into the Planck spectrum of blackbody radiation. | ||
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| ===== Inquiries ===== | ===== Inquiries ===== | ||
| * Derive the phase-space distribution \( f = g h^{-3} [e^{(\epsilon-\mu)/ | * Derive the phase-space distribution \( f = g h^{-3} [e^{(\epsilon-\mu)/ | ||
| - | * Show how the occupation number \( \bar{n}_i \) emerges from integrating \(f\) over a state’s phase-space cell. | + | * Show how the occupation number \( \bar{n}_i \) emerges from integrating \( f \) over a state’s phase-space cell. |
| * Demonstrate explicitly how the Maxwell–Boltzmann limit arises from the Bose–Einstein distribution. | * Demonstrate explicitly how the Maxwell–Boltzmann limit arises from the Bose–Einstein distribution. | ||
| - | * Using \( n\lambda_{\mathrm{th}}^{3} \sim 1 \), estimate the temperature at which a dilute Bose gas becomes quantum degenerate. | + | * Using \( n \lambda_{\mathrm{th}}^{3} \sim 1 \), estimate the temperature at which a dilute Bose gas becomes quantum degenerate. |
| - | * Explain how varying \(a=-\mu/ | + | * Explain how varying \( a = -\mu/(kT) \) alters the shape of the occupation curves in the interactive figure. |
un/bose-einstein-statistics.1764513201.txt.gz · Last modified: by asad