The modified lognormal power-law (MLP ) function is a three parameter function that can be used to model data that have characteristics of a log-normal distribution and a power law behavior. It has been used to model the functional form of the initial mass function (IMF). Unlike the other functional forms of the IMF, the MLP is a single function with no joining conditions.
The closed form of the probability density function of the MLP is as follows:
f ( m ) = α 2 exp ( α μ 0 + α 2 σ 0 2 2 ) m − ( 1 + α ) erfc ( 1 2 ( α σ 0 − ln ( m ) − μ 0 σ 0 ) ) , m ∈ [ 0 , ∞ ) {\displaystyle {\begin{aligned}f(m)={\frac {\alpha }{2}}\exp \left(\alpha \mu _{0}+{\frac {\alpha ^{2}\sigma _{0}^{2}}{2}}\right)m^{-(1+\alpha )}{\text{erfc}}\left({\frac {1}{\sqrt {2}}}\left(\alpha \sigma _{0}-{\frac {\ln(m)-\mu _{0}}{\sigma _{0}}}\right)\right),\ m\in [0,\infty )\end{aligned}}} where α = δ γ {\displaystyle {\begin{aligned}\alpha ={\frac {\delta }{\gamma }}\end{aligned}}} is the asymptotic power-law index of the distribution. Here μ 0 {\displaystyle \mu _{0}} and σ 0 2 {\displaystyle \sigma _{0}^{2}} are the mean and variance, respectively, of an underlying lognormal distribution from which the MLP is derived.
Mathematical properties [ edit ] Following are the few mathematical properties of the MLP distribution:
Cumulative distribution [ edit ] The MLP cumulative distribution function ( F ( m ) = ∫ − ∞ m f ( t ) d t {\displaystyle F(m)=\int _{-\infty }^{m}f(t)\,dt} ) is given by:
F ( m ) = 1 2 erfc ( − ln ( m ) − μ 0 2 σ 0 ) − 1 2 exp ( α μ 0 + α 2 σ 0 2 2 ) m − α erfc ( α σ 0 2 ( α σ 0 − ln ( m ) − μ 0 2 σ 0 ) ) {\displaystyle {\begin{aligned}F(m)={\frac {1}{2}}{\text{erfc}}\left(-{\frac {\ln(m)-\mu _{0}}{{\sqrt {2}}\sigma _{0}}}\right)-{\frac {1}{2}}\exp \left(\alpha \mu _{0}+{\frac {\alpha ^{2}\sigma _{0}^{2}}{2}}\right)m^{-\alpha }{\text{erfc}}\left({\frac {\alpha \sigma _{0}}{\sqrt {2}}}\left(\alpha \sigma _{0}-{\frac {\ln(m)-\mu _{0}}{{\sqrt {2}}\sigma _{0}}}\right)\right)\end{aligned}}} We can see that as m → 0 , {\displaystyle m\to 0,} that F ( m ) → 1 2 erfc ( − ln ( m − μ 0 ) 2 σ 0 ) , {\displaystyle \textstyle F(m)\to {\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln(m-\mu _{0})}{{\sqrt {2}}\sigma _{0}}}\right),} which is the cumulative distribution function for a lognormal distribution with parameters μ 0 and σ 0 .
Mean, variance, raw moments[ edit ] The expectation value of M {\displaystyle M} k gives the k {\displaystyle k} th raw moment of M {\displaystyle M} ,
⟨ M k ⟩ = ∫ 0 ∞ m k f ( m ) d m {\displaystyle {\begin{aligned}\langle M^{k}\rangle =\int _{0}^{\infty }m^{k}f(m)\mathrm {d} m\end{aligned}}} This exists if and only if α > k {\displaystyle k} , in which case it becomes:
⟨ M k ⟩ = α α − k exp ( σ 0 2 k 2 2 + μ 0 k ) , α > k {\displaystyle {\begin{aligned}\langle M^{k}\rangle ={\frac {\alpha }{\alpha -k}}\exp \left({\frac {\sigma _{0}^{2}k^{2}}{2}}+\mu _{0}k\right),\ \alpha >k\end{aligned}}} which is the k {\displaystyle k} th raw moment of the lognormal distribution with the parameters μ0 and σ0 scaled by α ⁄α- k {\displaystyle k} in the limit α→∞. This gives the mean and variance of the MLP distribution:
⟨ M ⟩ = α α − 1 exp ( σ 0 2 2 + μ 0 ) , α > 1 {\displaystyle {\begin{aligned}\langle M\rangle ={\frac {\alpha }{\alpha -1}}\exp \left({\frac {\sigma _{0}^{2}}{2}}+\mu _{0}\right),\ \alpha >1\end{aligned}}} ⟨ M 2 ⟩ = α α − 2 exp ( 2 ( σ 0 2 + μ 0 ) ) , α > 2 {\displaystyle {\begin{aligned}\langle M^{2}\rangle ={\frac {\alpha }{\alpha -2}}\exp \left(2\left(\sigma _{0}^{2}+\mu _{0}\right)\right),\ \alpha >2\end{aligned}}} Var( M {\displaystyle M} ) = ⟨ M {\displaystyle M} 2 ⟩-(⟨ M {\displaystyle M} ⟩)2 = α exp(σ0 2 + 2μ0 ) (exp(σ0 2 ) / α-2 - α / (α-2)2 ), α > 2
The solution to the equation f ′ ( m ) {\displaystyle f'(m)} = 0 (equating the slope to zero at the point of maxima) for m {\displaystyle m} gives the mode of the MLP distribution.
f ′ ( m ) = 0 ⇔ K erfc ( u ) = exp ( − u 2 ) , {\displaystyle f'(m)=0\Leftrightarrow K\operatorname {erfc} (u)=\exp(-u^{2}),} where u = 1 2 ( α σ 0 − ln m − μ 0 σ 0 ) {\displaystyle \textstyle u={\frac {1}{\sqrt {2}}}\left(\alpha \sigma _{0}-{\frac {\ln m-\mu _{0}}{\sigma _{0}}}\right)} and K = σ 0 ( α + 1 ) π 2 . {\displaystyle K=\sigma _{0}(\alpha +1){\tfrac {\sqrt {\pi }}{2}}.}
Numerical methods are required to solve this transcendental equation. However, noting that if K {\displaystyle K} ≈1 then u = 0 gives us the mode m {\displaystyle m} * :
m ∗ = exp ( μ 0 + α σ 0 2 ) {\displaystyle m^{*}=\exp(\mu _{0}+\alpha \sigma _{0}^{2})} The lognormal random variate is:
L ( μ , σ ) = exp ( μ + σ N ( 0 , 1 ) ) {\displaystyle {\begin{aligned}L(\mu ,\sigma )=\exp(\mu +\sigma N(0,1))\end{aligned}}} where N ( 0 , 1 ) {\displaystyle N(0,1)} is standard normal random variate. The exponential random variate is :
E ( δ ) = − δ − 1 ln ( R ( 0 , 1 ) ) {\displaystyle {\begin{aligned}E(\delta )=-\delta ^{-1}\ln(R(0,1))\end{aligned}}} where R(0,1) is the uniform random variate in the interval [0,1]. Using these two, we can derive the random variate for the MLP distribution to be:
M ( μ 0 , σ 0 , α ) = exp ( μ 0 + σ 0 N ( 0 , 1 ) − α − 1 ln ( R ( 0 , 1 ) ) ) {\displaystyle {\begin{aligned}M(\mu _{0},\sigma _{0},\alpha )=\exp(\mu _{0}+\sigma _{0}N(0,1)-\alpha ^{-1}\ln(R(0,1)))\end{aligned}}} Basu, Shantanu; Gil, M; Auddy, Sayatan (April 1, 2015). "The MLP distribution: a modified lognormal power-law model for the stellar initial mass function". MNRAS . 449 (3): 2413–2420. arXiv :1503.00023 . Bibcode :2015MNRAS.449.2413B . doi :10.1093/mnras/stv445 . {{cite journal }}
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