## Density of states → Helmholtz free energy density $f(T)$

Using statistical mechanics, it can be shown that the Helmholtz free energy density $f(T)$ for photons (where the chemical potential is zero $\mu=0$) can be expressed as the following integral,

$$f(T) = k_B T ~ \int\limits_0^{\infty}D(\omega) \ln \left[ 1- \exp \left( \frac{-\hbar\omega}{k_B T} \right) \right]d\omega.$$

Here $\hbar\omega$ is the energy of photons with frequency $\omega$ and $D(\omega)$ is the density of states. This result is discussed in Statistical Physics, Part 1 by Landau and Lifshitz and in the notes on the thermodynamic properties of bosons.

The form below uses this formula to calculate the temperature dependence of the Helmholtz free energy density from tabulated data for the density of states. The density of states data is input as two columns in the textbox at the lower left. The first column is the angular-frequency $\omega$ in rad/s. The second column is the density of states. The units of the density of states depends on the dimensionality: s/m for 1d, s/m² for 2d, and s/m³ for 3d.

After the 'DoS → f(T)' button is pressed, the density of states is plotted on the left and $f(T)$ is plotted from temperature $T_{\text{min}}$ to temperature $T_{\text{max}}$ on the right. The data for the $f(T)$ plot also appear in tabular form in the lower right textbox. The first column is the temperature in Kelvin and the second column is the Helmholtz free energy density in units of J m-1, J m-2, or J m-3 depending on the dimensionality.

 D(ω) f(T) ω [1015 rad/s] T [K] Tmin:  K  Tmax:  K Input: [ω D(ω)] Output: [T f(T)] 0 0 1E+13 0.376015807 2E+13 1.50406323 3E+13 3.384142267 4E+13 6.016252919 5E+13 9.400395186 6E+13 13.53656907 7E+13 18.42477456 8E+13 24.06501168 9E+13 30.4572804 1E+14 37.60158074 1.1E+14 45.4979127 1.2E+14 54.14627627 1.3E+14 63.54667146 1.4E+14 73.69909826 1.5E+14 84.60355667 1.6E+14 96.2600467 1.7E+14 108.6685683 1.8E+14 121.8291216 1.9E+14 135.7417065 2E+14 150.406323 2.1E+14 165.8229711 2.2E+14 181.9916508 2.3E+14 198.9123621 2.4E+14 216.5851051 2.5E+14 235.0098796 2.6E+14 254.1866858 2.7E+14 274.1155236 2.8E+14 294.796393 2.9E+14 316.2292941 3E+14 338.4142267 3.1E+14 361.3511909 3.2E+14 385.0401868 3.3E+14 409.4812143 3.4E+14 434.6742734 3.5E+14 460.6193641 3.6E+14 487.3164864 3.7E+14 514.7656404 3.8E+14 542.9668259 3.9E+14 571.9200431 4E+14 601.6252919 4.1E+14 632.0825723 4.2E+14 663.2918843 4.3E+14 695.2532279 4.4E+14 727.9666032 4.5E+14 761.4320101 4.6E+14 795.6494485 4.7E+14 830.6189186 4.8E+14 866.3404203 4.9E+14 902.8139537 5E+14 940.0395186 5.1E+14 978.0171151 5.2E+14 1016.746743 5.3E+14 1056.228403 5.4E+14 1096.462094 5.5E+14 1137.447817 5.6E+14 1179.185572 5.7E+14 1221.675358 5.8E+14 1264.917176 5.9E+14 1308.911026 6E+14 1353.656907 6.1E+14 1399.154819 6.2E+14 1445.404764 6.3E+14 1492.40674 6.4E+14 1540.160747 6.5E+14 1588.666786 6.6E+14 1637.924857 6.7E+14 1687.93496 6.8E+14 1738.697094 6.9E+14 1790.211259 7E+14 1842.477456 7.1E+14 1895.495685 7.2E+14 1949.265946 7.3E+14 2003.788238 7.4E+14 2059.062562 7.5E+14 2115.088917 7.6E+14 2171.867304 7.7E+14 2229.397722 7.8E+14 2287.680172 7.9E+14 2346.714654 8E+14 2406.501168 8.1E+14 2467.039713 8.2E+14 2528.330289 8.3E+14 2590.372897 8.4E+14 2653.167537 8.5E+14 2716.714209 8.6E+14 2781.012912 8.7E+14 2846.063646 8.8E+14 2911.866413 8.9E+14 2978.421211 9E+14 3045.72804 9.1E+14 3113.786901 9.2E+14 3182.597794 9.3E+14 3252.160719 9.4E+14 3322.475675 9.5E+14 3393.542662 9.6E+14 3465.361681 9.7E+14 3537.932732 9.8E+14 3611.255815 9.9E+14 3685.330929 1E+15 3760.158074 1.01E+15 3835.737252 1.02E+15 3912.068461 1.03E+15 3989.151701 1.04E+15 4066.986973 1.05E+15 4145.574277 1.06E+15 4224.913612 1.07E+15 4305.004979 1.08E+15 4385.848378 1.09E+15 4467.443808 1.1E+15 4549.79127 1.11E+15 4632.890763 1.12E+15 4716.742288 1.13E+15 4801.345845 1.14E+15 4886.701433 1.15E+15 4972.809053 1.16E+15 5059.668705 1.17E+15 5147.280388 1.18E+15 5235.644103 1.19E+15 5324.759849 1.2E+15 5414.627627 1.21E+15 5505.247437 1.22E+15 5596.619278 1.23E+15 5688.743151 1.24E+15 5781.619055 1.25E+15 5875.246991 1.26E+15 5969.626959 1.27E+15 6064.758958 1.28E+15 6160.642989 1.29E+15 6257.279052 1.3E+15 6354.667146 1.31E+15 6452.807271 1.32E+15 6551.699429 1.33E+15 6651.343618 1.34E+15 6751.739838 1.35E+15 6852.888091 1.36E+15 6954.788374 1.37E+15 7057.44069 1.38E+15 7160.845037 1.39E+15 7265.001415 1.4E+15 7369.909826 1.41E+15 7475.570268 1.42E+15 7581.982741 1.43E+15 7689.147246 1.44E+15 7797.063783 1.45E+15 7905.732351 1.46E+15 8015.152951 1.47E+15 8125.325583 1.48E+15 8236.250246 1.49E+15 8347.926941 1.5E+15 8460.355667 1.51E+15 8573.536425 1.52E+15 8687.469215 1.53E+15 8802.154036 1.54E+15 8917.590889 1.55E+15 9033.779774 1.56E+15 9150.72069 1.57E+15 9268.413637 1.58E+15 9386.858617 1.59E+15 9506.055628 1.6E+15 9626.00467 1.61E+15 9746.705745 1.62E+15 9868.15885 1.63E+15 9990.363988 1.64E+15 10113.32116 1.65E+15 10237.03036 1.66E+15 10361.49159 1.67E+15 10486.70485 1.68E+15 10612.67015 1.69E+15 10739.38748 1.7E+15 10866.85683 1.71E+15 10995.07823 1.72E+15 11124.05165 1.73E+15 11253.7771 1.74E+15 11384.25459 1.75E+15 11515.4841 1.76E+15 11647.46565 1.77E+15 11780.19923 1.78E+15 11913.68484 1.79E+15 12047.92249 1.8E+15 12182.91216 1.81E+15 12318.65387 1.82E+15 12455.14761 1.83E+15 12592.39338 1.84E+15 12730.39118 1.85E+15 12869.14101 1.86E+15 13008.64287 1.87E+15 13148.89677 1.88E+15 13289.9027 1.89E+15 13431.66066 1.9E+15 13574.17065 1.91E+15 13717.43267 1.92E+15 13861.44673 1.93E+15 14006.21281 1.94E+15 14151.73093 1.95E+15 14298.00108 1.96E+15 14445.02326 1.97E+15 14592.79747 1.98E+15 14741.32371 1.99E+15 14890.60199 2E+15 15040.6323 2.01E+15 15191.41464 2.02E+15 15342.94901 2.03E+15 15495.23541 2.04E+15 15648.27384 2.05E+15 15802.06431 2.06E+15 15956.6068 2.07E+15 16111.90133 2.08E+15 16267.94789 2.09E+15 16424.74648 2.1E+15 16582.29711 2.11E+15 16740.59976 2.12E+15 16899.65445 2.13E+15 17059.46117 2.14E+15 17220.01992 2.15E+15 17381.3307 2.16E+15 17543.39351 2.17E+15 17706.20836 2.18E+15 17869.77523 2.19E+15 18034.09414 2.2E+15 18199.16508 2.21E+15 18364.98805 2.22E+15 18531.56305 2.23E+15 18698.89009 2.24E+15 18866.96915 2.25E+15 19035.80025 2.26E+15 19205.38338 2.27E+15 19375.71854 2.28E+15 19546.80573 2.29E+15 19718.64496 2.3E+15 19891.23621 2.31E+15 20064.5795 2.32E+15 20238.67482 2.33E+15 20413.52217 2.34E+15 20589.12155 2.35E+15 20765.47297 2.36E+15 20942.57641 2.37E+15 21120.43189 2.38E+15 21299.0394 2.39E+15 21478.39894 2.4E+15 21658.51051 2.41E+15 21839.37411 2.42E+15 22020.98975 2.43E+15 22203.35741 2.44E+15 22386.47711 2.45E+15 22570.34884 2.46E+15 22754.9726 2.47E+15 22940.3484 2.48E+15 23126.47622 2.49E+15 23313.35608 2.5E+15 23500.98796 2.51E+15 23689.37188 2.52E+15 23878.50784 2.53E+15 24068.39582 2.54E+15 24259.03583 2.55E+15 24450.42788 2.56E+15 24642.57196 2.57E+15 24835.46807 2.58E+15 25029.11621 2.59E+15 25223.51638 2.6E+15 25418.66858 2.61E+15 25614.57282 2.62E+15 25811.22909 2.63E+15 26008.63738 2.64E+15 26206.79772 2.65E+15 26405.71008 2.66E+15 26605.37447 2.67E+15 26805.7909 2.68E+15 27006.95935 2.69E+15 27208.87984 2.7E+15 27411.55236 2.71E+15 27614.97691 2.72E+15 27819.1535 2.73E+15 28024.08211 2.74E+15 28229.76276 2.75E+15 28436.19544 2.76E+15 28643.38015 2.77E+15 28851.31689 2.78E+15 29060.00566 2.79E+15 29269.44647 2.8E+15 29479.6393 2.81E+15 29690.58417 2.82E+15 29902.28107 2.83E+15 30114.73 2.84E+15 30327.93096 2.85E+15 30541.88396 2.86E+15 30756.58899 2.87E+15 30972.04604 2.88E+15 31188.25513 2.89E+15 31405.21625 2.9E+15 31622.92941 2.91E+15 31841.39459 2.92E+15 32060.61181 2.93E+15 32280.58105 2.94E+15 32501.30233 2.95E+15 32722.77564 2.96E+15 32945.00098 2.97E+15 33167.97836 2.98E+15 33391.70776 2.99E+15 33616.1892 3E+15 33841.42267 3.01E+15 34067.40817 3.02E+15 34294.1457 3.03E+15 34521.63526 3.04E+15 34749.87686 3.05E+15 34978.87049 3.06E+15 35208.61615 3.07E+15 35439.11384 3.08E+15 35670.36356 3.09E+15 35902.36531 3.1E+15 36135.11909 3.11E+15 36368.62491 3.12E+15 36602.88276 3.13E+15 36837.89264 3.14E+15 37073.65455 3.15E+15 37310.16849 3.16E+15 37547.43447 3.17E+15 37785.45247 3.18E+15 38024.22251 3.19E+15 38263.74458 3.2E+15 38504.01868 3.21E+15 38745.04481 3.22E+15 38986.82298 3.23E+15 39229.35317 3.24E+15 39472.6354 3.25E+15 39716.66966 3.26E+15 39961.45595 3.27E+15 40206.99427 3.28E+15 40453.28463 3.29E+15 40700.32701 3.3E+15 40948.12143 3.31E+15 41196.66788 3.32E+15 41445.96636 3.33E+15 41696.01687 3.34E+15 41946.81941 3.35E+15 42198.37399 3.36E+15 42450.6806 3.37E+15 42703.73923 3.38E+15 42957.5499 3.39E+15 43212.11261 3.4E+15 43467.42734 3.41E+15 43723.4941 3.42E+15 43980.3129 3.43E+15 44237.88373 3.44E+15 44496.20659 3.45E+15 44755.28148 3.46E+15 45015.1084 3.47E+15 45275.68736 3.48E+15 45537.01834 3.49E+15 45799.10136 3.5E+15 46061.93641 3.51E+15 46325.52349 3.52E+15 46589.8626 3.53E+15 46854.95375 3.54E+15 47120.79692 3.55E+15 47387.39213 3.56E+15 47654.73937 3.57E+15 47922.83864 3.58E+15 48191.68994 3.59E+15 48461.29328 3.6E+15 48731.64864 3.61E+15 49002.75604 3.62E+15 49274.61547 3.63E+15 49547.22693 3.64E+15 49820.59042 3.65E+15 50094.70595 3.66E+15 50369.5735 3.67E+15 50645.19309 3.68E+15 50921.56471 3.69E+15 51198.68836 3.7E+15 51476.56404 3.71E+15 51755.19175 3.72E+15 52034.5715 3.73E+15 52314.70327 3.74E+15 52595.58708 3.75E+15 52877.22292 3.76E+15 53159.61079 3.77E+15 53442.7507 3.78E+15 53726.64263 3.79E+15 54011.2866 3.8E+15 54296.68259 3.81E+15 54582.83062 3.82E+15 54869.73068 3.83E+15 55157.38278 3.84E+15 55445.7869 3.85E+15 55734.94306 3.86E+15 56024.85124 3.87E+15 56315.51146 3.88E+15 56606.92371 3.89E+15 56899.088 3.9E+15 57192.00431 3.91E+15 57485.67266 3.92E+15 57780.09303 3.93E+15 58075.26544 3.94E+15 58371.18988 3.95E+15 58667.86636 3.96E+15 58965.29486 3.97E+15 59263.47539 3.98E+15 59562.40796 3.99E+15 59862.09256 4E+15 60162.52919 4.01E+15 60463.71785 4.02E+15 60765.65854 4.03E+15 61068.35127 4.04E+15 61371.79603 4.05E+15 61675.99281 4.06E+15 61980.94163 4.07E+15 62286.64249 4.08E+15 62593.09537 4.09E+15 62900.30028 4.1E+15 63208.25723 4.11E+15 63516.96621 4.12E+15 63826.42722 4.13E+15 64136.64026 4.14E+15 64447.60533 4.15E+15 64759.32244 4.16E+15 65071.79157 4.17E+15 65385.01274 4.18E+15 65698.98594 4.19E+15 66013.71117 4.2E+15 66329.18843 4.21E+15 66645.41773 4.22E+15 66962.39905 4.23E+15 67280.13241 4.24E+15 67598.6178 4.25E+15 67917.85522 4.26E+15 68237.84467 4.27E+15 68558.58615 4.28E+15 68880.07967 4.29E+15 69202.32522 4.3E+15 69525.32279 4.31E+15 69849.07241 4.32E+15 70173.57405 4.33E+15 70498.82772 4.34E+15 70824.83343 4.35E+15 71151.59116 4.36E+15 71479.10093 4.37E+15 71807.36273 4.38E+15 72136.37656 4.39E+15 72466.14242 4.4E+15 72796.66032 4.41E+15 73127.93025 4.42E+15 73459.9522 4.43E+15 73792.72619 4.44E+15 74126.25221 4.45E+15 74460.53027 4.46E+15 74795.56035 4.47E+15 75131.34247 4.48E+15 75467.87662 4.49E+15 75805.16279 4.5E+15 76143.20101 4.51E+15 76481.99125 4.52E+15 76821.53352 4.53E+15 77161.82783 4.54E+15 77502.87417 4.55E+15 77844.67253 4.56E+15 78187.22293 4.57E+15 78530.52537 4.58E+15 78874.57983 4.59E+15 79219.38633 4.6E+15 79564.94485 4.61E+15 79911.25541 4.62E+15 80258.318 4.63E+15 80606.13262 4.64E+15 80954.69928 4.65E+15 81304.01796 4.66E+15 81654.08868 4.67E+15 82004.91143 4.68E+15 82356.48621 4.69E+15 82708.81302 4.7E+15 83061.89186 4.71E+15 83415.72274 4.72E+15 83770.30564 4.73E+15 84125.64058 4.74E+15 84481.72755 4.75E+15 84838.56655 4.76E+15 85196.15759 4.77E+15 85554.50065 4.78E+15 85913.59575 4.79E+15 86273.44287 4.8E+15 86634.04203 4.81E+15 86995.39322 4.82E+15 87357.49645 4.83E+15 87720.3517 4.84E+15 88083.95899 4.85E+15 88448.3183 4.86E+15 88813.42965 4.87E+15 89179.29303 4.88E+15 89545.90845 4.89E+15 89913.27589 4.9E+15 90281.39537 4.91E+15 90650.26687 4.92E+15 91019.89041 4.93E+15 91390.26598 4.94E+15 91761.39358 4.95E+15 92133.27322 4.96E+15 92505.90488 4.97E+15 92879.28858 4.98E+15 93253.42431 4.99E+15 93628.31207

Photons in vacuum: