- Email: [email protected]
A new approach to the generalization of Planck’s law of black-body radiation
Accepted Manuscript A new approach to the generalization of Planck’s law of black-body radiation Sreeja Loho Choudhury, R.K. Paul
PII: DOI: Reference:
S0003-4916(18)30167-2 https://doi.org/10.1016/j.aop.2018.06.004 YAPHY 67691
To appear in:
Annals of Physics
Received date : 19 January 2018 Accepted date : 16 June 2018 Please cite this article as: S.L. Choudhury, R.K. Paul, A new approach to the generalization of Planck’s law of black-body radiation, Annals of Physics (2018), https://doi.org/10.1016/j.aop.2018.06.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
HIGHLIGHTS:
A new approximation approach to the generalized Planck’s radiation law has been developed.
The average energy of radiation has been derived by introducing the nonextensive partition function in the statistical relation of internal energy.
The spectral energy density and spectral radiance have been computed.
The present approach has been compared with the earlier developed approximate schemes and the recently developed exact method.
The deviations of the present approach and other approximate schemes (AA and FA) from the exact result have been computed for q=0.99947.
*Manuscript Click here to view linked References
A New Approach to the Generalization of Planck’s Law of Black-body Radiation Sreeja Loho Choudhury1, R. K. Paul2 Department of Physics, Birla Institute of Technology, Mesra, Ranchi-835215, Jharkhand, India
Abstract In this study, Planck’s law of black-body radiation has been modified within the framework of nonextensive statistical mechanics. The average energy of radiation has been derived by introducing the nonextensive partition function in the statistical relation of internal energy. The spectral energy density and spectral radiance have also been computed. The derived expression has been compared with the earlier developed approximate schemes (i.e. asymptotic approach and factorization approach) and with the recently obtained exact result. We utilize the exact and approximate Stefan-Boltzmann laws in order to compare with the new approach introduced here. Keywords: nonextensive partition function; generalized Planck’s radiation law; spectral energy density; spectral radiance; Stefan-Boltzmann Law; nonextensive statistical mechanics.
1
Email address: [email protected]
2
Email address: [email protected]
1. Introduction The extensive (additive) Boltzmann-Gibbs thermostatistics fails to study nonextensive physical systems where long range interactions or long range microscopic memory is involved or the system evolves in a (multi)fractal space-time. Therefore, the standard statistical mechanics is not universal and is valid only for extensive systems. Constantino Tsallis has proposed a nonextensive entropy (called Tsallis entropy) [1], which is a generalization of the traditional Boltzmann-Gibbs or Shannon entropy. The generalized Tsallis entropy [1] is given by, (1) where W is the number of microstates of the system and q is the nonextensive entropic index. The ordinary Boltzmann Gibbs entropy is obtained in the limit q→1. The generalized statistical mechanics has successfully been applied to investigate physical systems which exhibit nonextensive features like stellar polytrops, Levy-like anomalous diffusions, two dimensional turbulence, cosmic background radiation, solar neutrino problem and many others. Within the framework of the nonextensive statistical mechanics, we have generalized the Planck’s law for the black-body radiation. Earlier, an attempt was made to generalize the Planck’s radiation law (known as the asymptotic approach) [2] for the explanation of the cosmic microwave background radiation [3] at a temperature of 2.725 K. Another attempt was made to generalize statistics of quantum and classical gases using the factorization approach [4]. There are some versions of generalized Planck’s law available in the existing literature [5] in this regard. There are also recent attempts to generalize the Planck’s radiation law using Kaniadakis approach [6, 7]. We have derived the average energy of radiation by introducing the nonextensive partition function [8] in the statistical relation of internal energy. From the average energy, we have computed the spectral energy density and spectral radiance of the emitted radiation. It has been seen that this expression of average energy recovers the original Planck’s relation for q=1 (extensive).
2. Method Let us consider a cube of side L having conducting walls which is filled with electromagnetic radiation in thermal equilibrium at a temperature T. The radiation emitted from a small hole in one of the walls will be characteristic of a perfect black-body. According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by: (2) where
, n=0,1,2,3,..... and partition function (3)
Ordinary statistical mechanics is derived by maximizing the Boltzmann Gibbs entropy given by: (4) which is subjected to constraints, whereas in case of nonextensive statistical mechanics the more general Tsallis entropy [1] given by: (5) is maximized. Here
are probabilities associated with the microstates of a physical system
and q is the nonextensive entropic index. The ordinary Boltzmann Gibbs entropy is obtained in the limit q→1. For some given set of probabilities probabilities
, one can proceed to another set of
given as: (6)
The probability , where
coming out after maximizing
under the energy constraint
are the energy levels of the microstates is:
(7)
where statistical mechanics and
is the partition function [8] in case of nonextensive , = Boltzmann constant.
If we consider =n then, (8) (9)
2.1 Calculation of Average energy The average energy in a mode can be expressed in terms of the partition function as: (10) Now
Let (11) Now, (12) From the definition of limit we have,
.
(13) as q→1, (1-q)→0, and
then
(14) Therefore,
(15)
From (10) we have, (16) Let
where
(17)
Therefore,
(18)
Now, (19)
(20)
Hence,
(21)
Therefore,
(22)
From (9) we have the average energy given as:
For q=1 (extensive), we have (23)
Neglecting the vacuum energy term
, (24)
2.2 Energy Density The energy density per unit frequency is given by: (25) Therefore,
(26)
Hence the generalized energy density per unit frequency is given by:
For q=1, (27)
For q=0.95,
(28)
For q=1.05,
(29)
The variation of energy density with frequency for different q values has been shown in the following figure 1 for the temperature T=2.725K.
FIGURE 1. The plot of energy density versus frequency for different q values at a temperature T= 2.725 K.
2.3 Spectral Radiance The spectral radiance is given by:
(30) We have, frequency of radiation Therefore, So we get, (31)
(32)
Therfore, for q=1 we have, (33)
For q=0.95,
(34)
For q=1.05,
(35)
Figure 2 is the plot of spectral radiance versus wavelength for a particular temperature say T=2.725K. It is clear that the peak amplitude gradually decreases as q is increased. The shift of peak wavelength on the right hand side (higher wavelengths) with the increase in q values is also observed.
FIGURE 2. The plot of spectral radiance versus wavelength for the entropy index q= 0.95, 1 and 1.05 at a temperature T= 2.725K.
3. Comparison with other approaches In order to compare this new approximation approach with other previously introduced approaches, we utilize the Stefan-Boltzmann law. For the comparison of exact and approximate Stefan-Boltzmann laws with the Stefan-Boltzmann law of our approach, we will choose a q-value which is in accordance with the black-body radiation. The exact and approximate Stefan-Boltzmann laws have already been compared in [9]. The possible qcorrection could be of the order of 10-4 or 10-5. Hence, we will use here the largest deviation predicted for the q-correction [10], which is
and that gives q=0.99947.
The Stefan-Boltzmann law for our approach can be derived using (26) and integrating the generalized energy density per unit frequency over
, which gives the generalized energy
density. The Stefan-Boltzmann law for our approach and for the q value of 0.99947 is given by, (36) The Stefan-Boltzmann law derived by using asymptotic approximation (AA) [10, 11] and the factorization approach (FA) [12] is given by, (37) where θ=40.018 for the AA and θ=62.215 for the FA. V is the volume. There is also an exact analysis of the black-body radiation within the q-framework given in [13]. This exact analysis leads to the generalization of the Stefan-Boltzmann law as
(38)
where
(39) and (40)
Figure 3 shows the behaviour of our result (36), the approximate results (37) and the exact result (38) for q=0.99947. For q=1 (standard case), we use the original Stefan-Boltzmann law.
FIGURE 3. Black-body radiation: Internal energy versus (2πV1/3kT)/(hc) for q=0.99947. On zooming in the plot of internal energy, we can see that for such order of q-correction, the approximate results are closer to the standard (q=1) case, as well as to the exact result.
FIGURE 4. Black-body radiation: Internal energy versus (2πV1/3kT)/(hc) for q=0.99947 (Zoomed in).
4. Conclusion A new method of finding the average energy of a black-body is thus derived using the nonextensive statistical mechanics formalism. The spectral density and spectral radiance has also been derived following the average energy for entropy index q= 0.95, 1 and 1.05. For q=1, the expression of average energy recovers the original form of average energy in case of standard statistical mechanics (extensive). Hence this new approach to the generalization of Planck’s radiation law is justifiable and reliable. Several approaches like AA, FA and exact method have been employed earlier to study black-body radiation within the framework of nonextensive statistical mechanics. For the comparison of our approach with other previously developed approaches figures 3 and 4 have been plotted for internal energy of the black-body radiation. It seems from the figures that the
AA, FA and the present approach are in agreement with the exact result. However, for the sake of comparison, we calculate the deviations of approximate approaches from the exact result for q=0.99947. The formula for standard deviation is given by,
The above formula gives the deviation of each approximate approach from the exact approach for q=0.99947. The estimated deviation for different approaches is tabulated below. Method
Deviation
Our approach
1153.7
AA
796.6
FA
642.8 Table 1. Deviation from the exact result.
This implies that one can still rely on this new approximation method although there is an exact method. In a recent work [14] on Bose-Einstein and Fermi-Dirac distributions in nonextensive statistical mechanics, the results of the interpolation and superstatistical approaches are also in agreement with the exact approach within O(q-1). Hence the present approximation method gives a simple black-body radiation formula for the nonextensive case.
Acknowledgements The authors are grateful to the Physics department of Birla Institute of Technology, Mesra, Ranchi for rendering an outstanding environment to carry out the research work. The authors also acknowledge the help and support from M. K. Sinha (Physics Department, Birla Institute of Technology, Mesra, Ranchi).
References [1] C. Tsallis: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479487 (1988) [2] C. Tsallis, F. C. Sa Barreto and E. D. Loh: Generalization of the Planck radiation law and application to the cosmic microwave background radiation. Phys. Rev. E 52, 1447-1451 (1995) [3] J. C. Mather, et al.: Measurement of the cosmic microwve background spectrum by the COBE FIRAS instrument. Astrophys. J. 420, 439-444 (1994) [4] F. Büyükkiliç, D. Demirhan and A. Güleç: A statistical mechanical approach to generalized statistics of quantum and classical gases. Phys. Lett. A 197, 209-220 (1995) [5] U. Tirnakli, F. Büyükkiliç and D. Demirhan: Generalized Distribution Functions and an Alternative Approach to Generalized Planck Radiation Law. Physica A 240, 657-664 (1997) [6] K. Ourabah and M. Tribeche: Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics. Phys. Rev. E 89, 062130 (2014) [7] I. Lourek and M. Tribeche: Thermodynamic properties of the blackbody radiation. Phys. Lett. A 381, 452-456 (2017) [8] C. Beck: Non-extensive statistical mechanics and particle spectra in elementary interactions. Physica A. 286, 164-180 (2000) [9] U. Tirnakli and D. F. Torres: Exact and approximate results of non-extensive quantum statistics. Eur. Phys. J. B 14, 691-698 (2000) [10] A. R. Plastino, A. Plastino and H. Vucetich: A quantitative test of Gibbs’ statistical mechanics. Phys. Lett. A 207, 42-46 (1995) [11] Q. A. Wang, A. Le Méhauté: On the generalized blackbody distribution. Phys. Lett. A 237, 28-32 (1997) [12] U. Tirnakli, F. Büyükkiliç and D. Demirhan: Some bounds upon the nonextensivity parameter using the approximate generalized distribution functions. Phys. Lett. A 245, 62-66 (1998)
[13] E. K. Lenzi and R. S. Mendes: Blackbody radiation in nonextensive Tsallis statistics: Exact Solution. Phys. Lett. A 250, 270-274 (1998) [14] H. Hasegawa: Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: Exact and interpolation approaches. Phys. Rev. E 80, 011126 (2009)
PII: DOI: Reference:
S0003-4916(18)30167-2 https://doi.org/10.1016/j.aop.2018.06.004 YAPHY 67691
To appear in:
Annals of Physics
Received date : 19 January 2018 Accepted date : 16 June 2018 Please cite this article as: S.L. Choudhury, R.K. Paul, A new approach to the generalization of Planck’s law of black-body radiation, Annals of Physics (2018), https://doi.org/10.1016/j.aop.2018.06.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
HIGHLIGHTS:
A new approximation approach to the generalized Planck’s radiation law has been developed.
The average energy of radiation has been derived by introducing the nonextensive partition function in the statistical relation of internal energy.
The spectral energy density and spectral radiance have been computed.
The present approach has been compared with the earlier developed approximate schemes and the recently developed exact method.
The deviations of the present approach and other approximate schemes (AA and FA) from the exact result have been computed for q=0.99947.
*Manuscript Click here to view linked References
A New Approach to the Generalization of Planck’s Law of Black-body Radiation Sreeja Loho Choudhury1, R. K. Paul2 Department of Physics, Birla Institute of Technology, Mesra, Ranchi-835215, Jharkhand, India
Abstract In this study, Planck’s law of black-body radiation has been modified within the framework of nonextensive statistical mechanics. The average energy of radiation has been derived by introducing the nonextensive partition function in the statistical relation of internal energy. The spectral energy density and spectral radiance have also been computed. The derived expression has been compared with the earlier developed approximate schemes (i.e. asymptotic approach and factorization approach) and with the recently obtained exact result. We utilize the exact and approximate Stefan-Boltzmann laws in order to compare with the new approach introduced here. Keywords: nonextensive partition function; generalized Planck’s radiation law; spectral energy density; spectral radiance; Stefan-Boltzmann Law; nonextensive statistical mechanics.
1
Email address: [email protected]
2
Email address: [email protected]
1. Introduction The extensive (additive) Boltzmann-Gibbs thermostatistics fails to study nonextensive physical systems where long range interactions or long range microscopic memory is involved or the system evolves in a (multi)fractal space-time. Therefore, the standard statistical mechanics is not universal and is valid only for extensive systems. Constantino Tsallis has proposed a nonextensive entropy (called Tsallis entropy) [1], which is a generalization of the traditional Boltzmann-Gibbs or Shannon entropy. The generalized Tsallis entropy [1] is given by, (1) where W is the number of microstates of the system and q is the nonextensive entropic index. The ordinary Boltzmann Gibbs entropy is obtained in the limit q→1. The generalized statistical mechanics has successfully been applied to investigate physical systems which exhibit nonextensive features like stellar polytrops, Levy-like anomalous diffusions, two dimensional turbulence, cosmic background radiation, solar neutrino problem and many others. Within the framework of the nonextensive statistical mechanics, we have generalized the Planck’s law for the black-body radiation. Earlier, an attempt was made to generalize the Planck’s radiation law (known as the asymptotic approach) [2] for the explanation of the cosmic microwave background radiation [3] at a temperature of 2.725 K. Another attempt was made to generalize statistics of quantum and classical gases using the factorization approach [4]. There are some versions of generalized Planck’s law available in the existing literature [5] in this regard. There are also recent attempts to generalize the Planck’s radiation law using Kaniadakis approach [6, 7]. We have derived the average energy of radiation by introducing the nonextensive partition function [8] in the statistical relation of internal energy. From the average energy, we have computed the spectral energy density and spectral radiance of the emitted radiation. It has been seen that this expression of average energy recovers the original Planck’s relation for q=1 (extensive).
2. Method Let us consider a cube of side L having conducting walls which is filled with electromagnetic radiation in thermal equilibrium at a temperature T. The radiation emitted from a small hole in one of the walls will be characteristic of a perfect black-body. According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by: (2) where
, n=0,1,2,3,..... and partition function (3)
Ordinary statistical mechanics is derived by maximizing the Boltzmann Gibbs entropy given by: (4) which is subjected to constraints, whereas in case of nonextensive statistical mechanics the more general Tsallis entropy [1] given by: (5) is maximized. Here
are probabilities associated with the microstates of a physical system
and q is the nonextensive entropic index. The ordinary Boltzmann Gibbs entropy is obtained in the limit q→1. For some given set of probabilities probabilities
, one can proceed to another set of
given as: (6)
The probability , where
coming out after maximizing
under the energy constraint
are the energy levels of the microstates is:
(7)
where statistical mechanics and
is the partition function [8] in case of nonextensive , = Boltzmann constant.
If we consider =n then, (8) (9)
2.1 Calculation of Average energy The average energy in a mode can be expressed in terms of the partition function as: (10) Now
Let (11) Now, (12) From the definition of limit we have,
.
(13) as q→1, (1-q)→0, and
then
(14) Therefore,
(15)
From (10) we have, (16) Let
where
(17)
Therefore,
(18)
Now, (19)
(20)
Hence,
(21)
Therefore,
(22)
From (9) we have the average energy given as:
For q=1 (extensive), we have (23)
Neglecting the vacuum energy term
, (24)
2.2 Energy Density The energy density per unit frequency is given by: (25) Therefore,
(26)
Hence the generalized energy density per unit frequency is given by:
For q=1, (27)
For q=0.95,
(28)
For q=1.05,
(29)
The variation of energy density with frequency for different q values has been shown in the following figure 1 for the temperature T=2.725K.
FIGURE 1. The plot of energy density versus frequency for different q values at a temperature T= 2.725 K.
2.3 Spectral Radiance The spectral radiance is given by:
(30) We have, frequency of radiation Therefore, So we get, (31)
(32)
Therfore, for q=1 we have, (33)
For q=0.95,
(34)
For q=1.05,
(35)
Figure 2 is the plot of spectral radiance versus wavelength for a particular temperature say T=2.725K. It is clear that the peak amplitude gradually decreases as q is increased. The shift of peak wavelength on the right hand side (higher wavelengths) with the increase in q values is also observed.
FIGURE 2. The plot of spectral radiance versus wavelength for the entropy index q= 0.95, 1 and 1.05 at a temperature T= 2.725K.
3. Comparison with other approaches In order to compare this new approximation approach with other previously introduced approaches, we utilize the Stefan-Boltzmann law. For the comparison of exact and approximate Stefan-Boltzmann laws with the Stefan-Boltzmann law of our approach, we will choose a q-value which is in accordance with the black-body radiation. The exact and approximate Stefan-Boltzmann laws have already been compared in [9]. The possible qcorrection could be of the order of 10-4 or 10-5. Hence, we will use here the largest deviation predicted for the q-correction [10], which is
and that gives q=0.99947.
The Stefan-Boltzmann law for our approach can be derived using (26) and integrating the generalized energy density per unit frequency over
, which gives the generalized energy
density. The Stefan-Boltzmann law for our approach and for the q value of 0.99947 is given by, (36) The Stefan-Boltzmann law derived by using asymptotic approximation (AA) [10, 11] and the factorization approach (FA) [12] is given by, (37) where θ=40.018 for the AA and θ=62.215 for the FA. V is the volume. There is also an exact analysis of the black-body radiation within the q-framework given in [13]. This exact analysis leads to the generalization of the Stefan-Boltzmann law as
(38)
where
(39) and (40)
Figure 3 shows the behaviour of our result (36), the approximate results (37) and the exact result (38) for q=0.99947. For q=1 (standard case), we use the original Stefan-Boltzmann law.
FIGURE 3. Black-body radiation: Internal energy versus (2πV1/3kT)/(hc) for q=0.99947. On zooming in the plot of internal energy, we can see that for such order of q-correction, the approximate results are closer to the standard (q=1) case, as well as to the exact result.
FIGURE 4. Black-body radiation: Internal energy versus (2πV1/3kT)/(hc) for q=0.99947 (Zoomed in).
4. Conclusion A new method of finding the average energy of a black-body is thus derived using the nonextensive statistical mechanics formalism. The spectral density and spectral radiance has also been derived following the average energy for entropy index q= 0.95, 1 and 1.05. For q=1, the expression of average energy recovers the original form of average energy in case of standard statistical mechanics (extensive). Hence this new approach to the generalization of Planck’s radiation law is justifiable and reliable. Several approaches like AA, FA and exact method have been employed earlier to study black-body radiation within the framework of nonextensive statistical mechanics. For the comparison of our approach with other previously developed approaches figures 3 and 4 have been plotted for internal energy of the black-body radiation. It seems from the figures that the
AA, FA and the present approach are in agreement with the exact result. However, for the sake of comparison, we calculate the deviations of approximate approaches from the exact result for q=0.99947. The formula for standard deviation is given by,
The above formula gives the deviation of each approximate approach from the exact approach for q=0.99947. The estimated deviation for different approaches is tabulated below. Method
Deviation
Our approach
1153.7
AA
796.6
FA
642.8 Table 1. Deviation from the exact result.
This implies that one can still rely on this new approximation method although there is an exact method. In a recent work [14] on Bose-Einstein and Fermi-Dirac distributions in nonextensive statistical mechanics, the results of the interpolation and superstatistical approaches are also in agreement with the exact approach within O(q-1). Hence the present approximation method gives a simple black-body radiation formula for the nonextensive case.
Acknowledgements The authors are grateful to the Physics department of Birla Institute of Technology, Mesra, Ranchi for rendering an outstanding environment to carry out the research work. The authors also acknowledge the help and support from M. K. Sinha (Physics Department, Birla Institute of Technology, Mesra, Ranchi).
References [1] C. Tsallis: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479487 (1988) [2] C. Tsallis, F. C. Sa Barreto and E. D. Loh: Generalization of the Planck radiation law and application to the cosmic microwave background radiation. Phys. Rev. E 52, 1447-1451 (1995) [3] J. C. Mather, et al.: Measurement of the cosmic microwve background spectrum by the COBE FIRAS instrument. Astrophys. J. 420, 439-444 (1994) [4] F. Büyükkiliç, D. Demirhan and A. Güleç: A statistical mechanical approach to generalized statistics of quantum and classical gases. Phys. Lett. A 197, 209-220 (1995) [5] U. Tirnakli, F. Büyükkiliç and D. Demirhan: Generalized Distribution Functions and an Alternative Approach to Generalized Planck Radiation Law. Physica A 240, 657-664 (1997) [6] K. Ourabah and M. Tribeche: Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics. Phys. Rev. E 89, 062130 (2014) [7] I. Lourek and M. Tribeche: Thermodynamic properties of the blackbody radiation. Phys. Lett. A 381, 452-456 (2017) [8] C. Beck: Non-extensive statistical mechanics and particle spectra in elementary interactions. Physica A. 286, 164-180 (2000) [9] U. Tirnakli and D. F. Torres: Exact and approximate results of non-extensive quantum statistics. Eur. Phys. J. B 14, 691-698 (2000) [10] A. R. Plastino, A. Plastino and H. Vucetich: A quantitative test of Gibbs’ statistical mechanics. Phys. Lett. A 207, 42-46 (1995) [11] Q. A. Wang, A. Le Méhauté: On the generalized blackbody distribution. Phys. Lett. A 237, 28-32 (1997) [12] U. Tirnakli, F. Büyükkiliç and D. Demirhan: Some bounds upon the nonextensivity parameter using the approximate generalized distribution functions. Phys. Lett. A 245, 62-66 (1998)
[13] E. K. Lenzi and R. S. Mendes: Blackbody radiation in nonextensive Tsallis statistics: Exact Solution. Phys. Lett. A 250, 270-274 (1998) [14] H. Hasegawa: Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: Exact and interpolation approaches. Phys. Rev. E 80, 011126 (2009)