Discrete Kernel Preserving Model for 3D Electron–Optical Phonon Scattering Under Arbitrary Band Structures

Abstract

In Li et al. (J Sci Comput 62:317–335, 2015), we thoroughly investigated the structure of the kernel space of the discrete one-dimensional (1D) non-polar optical phonon (NPOP)-electron scattering matrix, and proposed a strategy to setup grid points so that the uniqueness of the discrete scattering kernel is preserved. In this paper, we extend the above work to the three dimensional (3D) case, and also investigate the polar optical phonon (POP)-electron case. In numerical discretization, it is important to get a discretization scattering matrix that keeps as many properties of the continuous scattering operator as possible. We prove that for the 3D NPOP-electron scattering, (i) the dimension of the kernel space of the discrete scattering matrix is one and (ii) the equilibrium distribution is constant with respect to the angular coordinates as long as the mesh over the energy interval obeys the rule proposed in Li et al. (2015). For the POP-electron scattering, (i) is also kept under the same condition, but to keep (ii) becomes a challenging task, since generally a simple uniform mesh in the azimuth and polar angular coordinates will not preserve (ii). Based on high degree of symmetry of the Platonic solids and regular pyramids, we propose two conditions and prove they are sufficient to guarantee (ii). Numerical experiments strongly support our theoretical findings.

Appendices

A Main Results Retrieved from [8]

In this section, we organize main results retrieved from our previous paper [8], which are used frequently to support our proofs in this paper.

Lemma 5

Linear ODEs system

$$\begin{aligned} \dfrac{\,\mathrm {d}{{{\varvec{g}}}}}{\,\mathrm {d}{t}} = M {{\varvec{g}}}, M \in {\mathbb {R}}^{N\times N}, \end{aligned}$$

(38)

satisfies the following (Q1)–(Q5), only if M satisfies all the conditions listed in Table 4.

• (Q1)
  $$\begin{aligned} \text {rank}(M) = N-1. \end{aligned}$$

• (Q2) Let ${{\varvec{g}}}\ne {\varvec{0}}$, and at least one of the entries of ${{\varvec{g}}}$ is positive, satisfy

  $$\begin{aligned} M{{\varvec{g}}}= {\varvec{0}}, \end{aligned}$$

  then the entries of ${{\varvec{g}}}$ are all positive.

• (Q3) Every nonzero eigenvalue of M has a negative real part.

• (Q4) $\lambda = 0$ is a semisimple eigenvalue of M.

• (Q5) For arbitrary ${{\varvec{g}}}(0) \ne {\varvec{0}}$, there exists a unique ${\varvec{g}} \in \mathrm {Ker}(M)$ such that the solutions of (38) satisfying

  $$\begin{aligned} \parallel {{\varvec{g}}}(t) - {{\varvec{g}}}\parallel _{l^2} \le \exp (-Ct)\parallel {{\varvec{g}}}(0) - {{\varvec{g}}}\parallel _{l^2}, \quad C>0, \end{aligned}$$

  and ${{\varvec{g}}}$ satisfies ${{\varvec{g}}}(0)-{{\varvec{g}}}\in \mathrm {Ker}^{\perp }(M)$.

Readers interested in the proof of Lemma 5 are recommended to refer to the proof of Theorem 1, Lemma 3, Lemma 1, Theorem 2 and Theorem 3, respectively, of our previous paper [8].

Table 4 The coefficient matrix M of (38)

If M is reducible, we could use transformations ${\mathbb {T}}_P$ and ${\mathbb {V}}_P$ to reformulate M and ${{\varvec{g}}}$ as

$$\begin{aligned} {\mathbb {T}}_P(M) = P^T M P = \left[ \begin{array}{ccc} M_1 &{} &{} \\ &{} \ddots &{} \\ &{} &{} M_s \end{array}\right] ,\quad {\mathbb {V}}_P({{\varvec{g}}}) = P {{\varvec{g}}}= \left( \begin{array}{c} {{\varvec{g}}}_1 \\ \vdots \\ {{\varvec{g}}}_s \end{array} \right) , \end{aligned}$$

(39)

where P is a permutation matrix. Apparently, if M still satisfies the last three conditions listed in Table 4, then we could find a permutation matrix P such that every diagonal-block $M_i$ of ${\mathbb {T}}_P(M)$ satisfies all the conditions in Table 4. As a result, any of the ODEs systems

$$\begin{aligned} \dfrac{\,\mathrm {d}{{{\varvec{g}}}_i}}{\,\mathrm {d}{t}} = M_i {{\varvec{g}}}_i, \quad \forall i = 1,\ldots ,s \end{aligned}$$

satisfies Lemma 5.

When M is reducible, the following Lemma 6 is obviously established.

Lemma 6

If M is reducible but satisfies all the other three conditions in Table 4, the M can also be reformulated as (39) with a proper ${\mathbb {T}}_P$, where

$$\begin{aligned} \mathrm {rank}(M) = N - s, \quad \mathrm {Dim}(\mathrm {Ker}(M))= s. \end{aligned}$$

In addition

$$\begin{aligned} \mathrm {Ker}(M) = \left\{ c_1\left( \begin{array}{c} {{\varvec{g}}}_1 \\ {\varvec{0}} \\ \vdots \\ {\varvec{0}} \end{array}\right) + c_2\left( \begin{array}{c} {\varvec{0}} \\ {{\varvec{g}}}_2 \\ \vdots \\ {\varvec{0}} \end{array}\right) + c_s\left( \begin{array}{c} {\varvec{0}} \\ {\varvec{0}} \\ \vdots \\ {{\varvec{g}}}_s \end{array}\right) :\quad c_i > 0,\quad {{\varvec{g}}}_i \in \mathrm {Ker}(M_i) \right\} . \end{aligned}$$

(40)

Specifically, the discrete 1D scattering system discussed in [8] has the form of (38), where

$$\begin{aligned} \begin{array}{ccc} M = (-2\varLambda + S),&\quad \varLambda = \mathrm {diag}\{ \lambda _1,\ldots ,\lambda _N\},&\quad S = \{s_{ij}\}_{i,j=1,\ldots ,N}, \end{array} \end{aligned}$$

(41)

in which

$$\begin{aligned}\begin{aligned} s_{ij}&= \frac{1}{\varDelta k_j} \left( \int _{I_{i}(E-\epsilon _p)\cap I_j(E)} \frac{s_1^{\mathrm {npo}}(E,E+\epsilon _p)}{\sqrt{2(E+\epsilon _p)}\sqrt{2E}} \,\mathrm {d}E + \int _{I_{i}(E+\epsilon _p)\cap I_j(E)} \frac{s_{-1}^{\mathrm {npo}}(E,E-\epsilon _p)}{\sqrt{2(E-\epsilon _p)}\sqrt{2E}} \,\mathrm {d}E\right) \\ \lambda _i&= \frac{1}{\varDelta k_i} \int _{I_i(E)} \left( \frac{s_{-1}^{\mathrm {npo}}(E,E-\epsilon _p)}{\sqrt{2(E-\epsilon _p)}\sqrt{2E}} + \frac{s_{1}^{\mathrm {npo}}(E,E+\epsilon _p)}{\sqrt{2(E+\epsilon _p)}\sqrt{2E}}\right) \,\mathrm {d}E. \end{aligned} \end{aligned}$$

The kernel space of M given above is s-dimensional and given as (40) if M is reducible and can be permuted into a block-diagonal matrix with s irreducible diagonal blocks.

Lemma 7 tells us the relation between the grid in energy and the reducibility of the scattering matrix M defined by (41), which is also the Theorem 4 firstly proposed in our previous paper [8].

Lemma 7

For $l > 1$, the following two items are equivalent:

• I: the scattering matrix M is reducible and can be permuted into a block-diagonal matrix with l diagonal blocks.

• II: $\exists E^\star _1,\ldots , E^\star _l \in [0, \epsilon _p)$, $E^\star _1< \cdots < E^\star _l$, such that

  $$\begin{aligned} \{ E^\star _i + \tau \epsilon _p : E^\star _i + \tau \epsilon _p \le E_{\max }, \tau \in {\mathbb {N}}\} \subseteq \{ E_{j - \frac{1}{2}} : j = 1, \ldots , N_E+1 \}, \end{aligned}$$

  (42)

  $i=1,\ldots ,l$.

In some special situations, the scattering matrix is irreducible for sure, such as

• 1. The scattering matrix M is irreducible if

  $$\begin{aligned} \text {max}_{1 \le j\le N_E} \varDelta E_j \ge \epsilon _p. \end{aligned}$$

• 2. For an equally distributed grid in energy, i.e., the spacing of arbitrary two neighbouring grid points in energy is $\varDelta E$, the scattering matrix M is irreducible if the ratio of $\epsilon _p$ and $\varDelta E$ is not an integer.

B Proof of Theorems and Lemmas in this paper

Proof of Theorem 1

Assume $f_h^{\mathrm s}(E,\varvec{\alpha }) = \sum _{i=1}^{N_E} C_i 1_i(E)$ such that, ${\hat{S}}^{\mathrm s}[f_h^{\mathrm s}] = 0$. Then as long as $C_i > 0$, $\forall i = 1,\ldots ,N_E$, is proved, the theorem is then concluded. We substitute $f_h^{\mathrm s}(E,\varvec{\alpha })$ in (22) with its above definition, then the equilibrium of (22) is obtained by solving the following linear equations:

$$\begin{aligned} G(\varvec{\alpha }) {\varvec{C}}= {\varvec{0}}, \quad \forall \varvec{\alpha }\in \varOmega , \end{aligned}$$

(43)

where

$$\begin{aligned} G(\varvec{\alpha }) = (g_{ij})_{N_E \times N_E}, ~ {\varvec{C}}= (C_1,C_2,\ldots ,C_{N_E})^T \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} g_{ij}&= \int _{I_i(E-\epsilon _p)\cap I_j(E)}\,\mathrm {d}\mu ^{E}C(E+\epsilon _p) \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'} {\bar{s}}_{1}(E,\varvec{\alpha }';E+\epsilon _p,\varvec{\alpha }) \\&\quad + \int _{I_i(E+\epsilon _p)\cap I_j(E)}\,\mathrm {d}\mu ^{E}C(E-\epsilon _p) \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'} {\bar{s}}_{-1}(E,\varvec{\alpha }';E-\epsilon _p,\varvec{\alpha }) \\&\quad - \delta _{ij}\int _{\varOmega } \,\mathrm {d}\mu ^{\varvec{\alpha }'} \int _{I_i(E)}\,\mathrm {d}\mu ^{E}[C(E-\epsilon _p) {\bar{s}}_{-1}(E,\varvec{\alpha };E-\epsilon _p,\varvec{\alpha }') \\&\quad + C(E+\epsilon _p) {\bar{s}}_{1}(E,\varvec{\alpha };E+\epsilon _p,\varvec{\alpha }')]. \end{aligned} \end{aligned}$$

(44)

According to the rotational invariance of the semi-discretized scattering operator, one see $G(\varvec{\alpha })$ does not depend on $\varvec{\alpha }$, i.e.,

$$\begin{aligned} G(\varvec{\alpha }) = G, \quad \forall \varvec{\alpha }\in \varOmega , \end{aligned}$$

thus the nontrivial solution ${\varvec{C}}$ of (43), if exists, does not dependent on $\varvec{\alpha }$, too.

To discuss the existence of nontrivial ${\varvec{C}}$ which satisfies (43), we make use of

$$\begin{aligned} {\bar{s}}_{\pm 1}(E_1,\varvec{\alpha }_1;E_2,\varvec{\alpha }_2) = {\bar{s}}_{\pm 1}(E_1,\varvec{\alpha }_2;E_2,\varvec{\alpha }_1) > 0, \quad \forall E_1,E_2\in [0,E_{\max }],~\varvec{\alpha }_1,\varvec{\alpha }_2\in \varOmega , \end{aligned}$$

and knows that G definitely satisfies all the conditions listed in Table 4, except the Irreducibility. There exists a permutation matrix P such that

$$\begin{aligned} \begin{array}{cc} {\mathbb {T}}_P(G) = \left[ \begin{array}{ccc} G_1 &{} &{} \\ &{} \ddots &{} \\ &{} &{} G_s \end{array}\right] , &{}\quad {\mathbb {V}}_P({\varvec{C}}) = \left[ \begin{array}{c}{\varvec{C}}_1\\ \vdots \\ {\varvec{C}}_s \end{array}\right] , \quad s\ge 1, \end{array} \end{aligned}$$

(45)

where the definitions of ${\mathbb {T}}_p$ and ${\mathbb {V}}_P$ are given in (39). Obviously, every $G_i$ satisfies all the conditions listed in Table 4, thus there exists ${\varvec{C}}_i > 0$( all the entries of ${\varvec{C}}$ are positive ) such that $G_i {\varvec{C}}_i = {\varvec{0}}$, for $i = 1,2,\ldots ,s$. $\square$

Proof of Theorem 2

Recalling (21) and (25), we multiply (22) with $\varPsi _j(\alpha )$, $j\in {\mathcal {J}}$, and calculate the Lebesgue integral of the resulted equation on $\varOmega$ with measure $\mu ^{\varvec{\alpha }}$, i.e.,

$$\begin{aligned}&V_i \dfrac{\,\mathrm {d}{f^{\mu _{\varvec{\alpha }}}_{ij}(t)}}{\,\mathrm {d}{t}} \nonumber \\&\quad = \left( \delta _{0j}\sum _{k}\frac{1}{V_k} \sum _{m}\delta _{0m}\left( V_k f^{\mu _{\varvec{\alpha }}}_{km}\right) \right) \left[ \int _{I_i(E-\epsilon _p)\cap I_k(E)}s_1^\mathrm{npo}(E,E+\epsilon _p) C(E+\epsilon _p)\,\mathrm {d}\mu ^E \right. \nonumber \\&\qquad \left. +\int _{I_i(E+\epsilon _p)\cap I_k(E)} s_{-1}^\mathrm{npo}(E,E-\epsilon _p)C(E-\epsilon _p)\,\mathrm {d}\mu ^E\right] \nonumber \\&\qquad - \left( \sum _{k}\frac{1}{V_i}\left( V_i f^{\mu _{\varvec{\alpha }}}_{ik}\right) \delta _{kj}\right) \int _{I_i(E)}\,\mathrm {d}\mu ^E [s_{-1}^\mathrm{npo}(E,E-\epsilon _p)C(E-\epsilon _p)\nonumber \\&\qquad + s_{1}^\mathrm{npo}(E,E+\epsilon _p)C(E+\epsilon _p)], \quad \text {for}~\, \forall i=1,\ldots ,N_E,~ j\in {\mathcal {J}}. \end{aligned}$$

(46)

Denoting by $F_{ij} = V_i f^{\mu _{\varvec{\alpha }}}_{ij}$ and ${\varvec{F}}_i = [F_{1i},\ldots ,F_{N_Ei}]^T$, we rewrite (46) into a series of matrix form ODEs:

$$\begin{aligned} \begin{array}{ccc} \dfrac{\,\mathrm {d}{{\varvec{F}}_i}}{\,\mathrm {d}{t}} = A_i{\varvec{F}}_i,~ A_i\in {\mathbb {R}}^{N_E\times N_E}, \quad i\in {\mathcal {J}}, \end{array} \end{aligned}$$

where $A_i$, $i=1,2,\ldots$, except $A_0$, are all diagonal matrices and negative definite. As a result,

$$\begin{aligned} F_{ji}(t) = F_{ji}(0)\exp \left( A_i(j,j) t\right) ,\quad \forall i>0,~ j=1,\ldots ,N_E, \end{aligned}$$

(47)

where $A_i(j,j)$ is the $j\hbox {th}$ diagonal entry of $A_i$. (47) implies when t goes to infinity, $F_{ji}(t)$ will decay to zero for all $i>0$ and $j=1,\ldots N_E$. In another word, if $f^s_h(E,\varvec{\alpha }) \in \mathrm {Ker}({\hat{S}}^s)$, then

$$\begin{aligned} f^s_h(E,\varvec{\alpha }) = g(E) = \sum _{i=1}^{N_E}\left( \frac{F_{i0}\varPsi _0}{V_i}\right) 1_i(E), \quad \varPsi _0 = \frac{1}{\sqrt{4\pi }}, \end{aligned}$$

(48)

where $F_{i0}$, $i=1,\ldots ,N_E$ in (48) are determined by

$$\begin{aligned} \dfrac{\,\mathrm {d}{{\varvec{F}}_0}}{\,\mathrm {d}{t}} = A_0 {\varvec{F}}_0, \quad {\varvec{F}}_0 = \left( \begin{array}{c} F_{10}\\ \vdots \\ F_{N_E 0} \end{array}\right) . \end{aligned}$$

(49)

$A_0$ in (49) is defined as

$$\begin{aligned} \begin{array}{ccc} A_0 = -\varLambda + S,&\varLambda = \mathrm {diag}\{ \lambda _1,\ldots ,\lambda _{N_E}\},&S = \{s_{ij}\}_{i,j=1,\ldots ,N_E}, \end{array} \end{aligned}$$

(50)

where

$$\begin{aligned} \begin{aligned} s_{ij}&= \frac{1}{V_j} \left[ \int _{I_{i}(E-\epsilon _p)\cap I_j(E)} s_1^{\mathrm {npo}}(E,E+\epsilon _p)C(E+\epsilon _p) \,\mathrm {d}\mu ^E \right. \\&\quad \left. + \int _{I_{i}(E+\epsilon _p)\cap I_j(E)} s_{-1}^{\mathrm {npo}}(E,E-\epsilon _p)C(E-\epsilon _p) \,\mathrm {d}\mu ^E\right] , \\ \lambda _i&= \frac{1}{V_i} \int _{I_i(E)} \,\mathrm {d}\mu ^E \left( s_{-1}^{\mathrm {npo}}(E,E-\epsilon _p)C(E-\epsilon _p) + s_{1}^{\mathrm {npo}}(E,E+\epsilon _p) C(E+\epsilon _p)\right) . \end{aligned} \end{aligned}$$

Clearly, $A_0$ defined in (50) satisfies the last three conditions stated in Table 4, which allows us to choose an appropriate permutation matrix P and make transformations

$$\begin{aligned} {\mathbb {T}}_P( A_0 ) = \left[ \begin{array}{ccc}M_1 &{}&{} \\ {} &{}\ddots &{} \\ &{}&{} M_s\end{array}\right] ,\quad {\mathbb {V}}_P({\varvec{F}}_0) = \left[ \begin{array}{c} {{\varvec{g}}}_1\\ \vdots \\ {{\varvec{g}}}_s\end{array}\right] , \end{aligned}$$

(51)

according to (39). Obviously, every $M_i$ satisfies all the conditions listed in Table 4, thus there exists unique ${{\varvec{g}}}_i > {\varvec{0}}$ such that $\mathrm {Ker}(M_i) = \left\{ c_i {{\varvec{g}}}_i: c_i > 0\right\}$ according to (Q5) stated in Lemma 5. Consequently, $\mathrm {Ker}(M)$ is nontrivial and defined as (40), which means $F_{i0} > 0$, $i=1,\ldots ,N_E$ in (48) are well defined. By denoting $C_i = \left( \frac{F_{i0}\varPsi _0}{V_i}\right)$, one sees that $\mathrm {Ker}({\hat{S}}^s) \in {\mathcal {L}}^{\mathrm {s}}$, and $\mathrm {Ker}({\hat{S}}^s) = {\mathcal {L}}^{\mathrm {s}}$ is thus concluded by noticing ${\mathcal {L}}^{\mathrm {s}} \in \mathrm {Ker}({\hat{S}}^{\mathrm {s}})$ is obvious according to the definition of ${\mathcal {L}}^{\mathrm {s}}$. $\square$

Proof of Lemma 1

Obviously, $\mathrm {Dim}( \mathrm {Ker}({\hat{S}}^s)) =\mathrm {Dim}(\mathrm {Ker}(A_0)) = s$ according to Lemma 6, where $A_0$ is defined in (50). Comparing entries of $A_0$ and M in (41), one notices that if the same grids in energy are used when constructing $A_0$ and M, then

$$\begin{aligned} A_0(i,j) \ne 0 \Leftrightarrow M(i,j) \ne 0, \quad \forall 1\le i,j\le N_E. \end{aligned}$$

Therefore, Lemma 7 is also suitable for the system (49) if we treat $A_0$ as the scattering matrix M as described in Lemma 7, which concludes the Lemma. $\square$

Proof of Lemma 2

Introducing nondecreasing functions $\chi :{\mathbb {R}}\rightarrow {\mathbb {R}}$, we have

$$\begin{aligned}&\int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }}\sum _{i=1}^{N_E} \left( {\hat{S}}^{\mathrm {s}}\left[ f^{\mathrm s}_h(t,E,\varvec{\alpha }) \chi \left( \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha })}{g(E)}\right) \right] \right) _{i}\nonumber \\&\quad = \sum _{i=1}^{N_E} \int _{I_i(E)}\,\mathrm {d}\mu ^{E}C(E-\epsilon _p) \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'}\nonumber \\&[s^{\mathrm {s}}_{1}(E-\epsilon _p,\varvec{\alpha }'; E,\varvec{\alpha }) f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha }') -s^{\mathrm {s}}_{-1}(E,\varvec{\alpha };E-\epsilon _p,\varvec{\alpha }') f^{\mathrm s}_h(t,E,\varvec{\alpha })]\chi \left( \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha })}{g(E)}\right) \nonumber \\&\qquad + \sum _{i=1}^{N_E}\int _{I_i(E)}\,\mathrm {d}\mu ^{E} C(E+\epsilon _p) \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'} \nonumber \\&[s^{\mathrm {s}}_{-1}(E+\epsilon _p,\varvec{\alpha }';E,\varvec{\alpha }) f^{\mathrm s}_h(t,E+\epsilon _p,\varvec{\alpha }') -s^{\mathrm {s}}_{1}(E,\varvec{\alpha };E+\epsilon _p,\varvec{\alpha }') f^{\mathrm s}_h(t,E,\varvec{\alpha })]\chi \left( \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha })}{g(E)}\right) \nonumber \\&\quad = \sum _{i=1}^{N_E} \int _{I_i(E)}C(E-\epsilon _p)\,\mathrm {d}\mu ^{E} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'}\nonumber \\&[s^{\mathrm {s}}_{1}(E-\epsilon _p,\varvec{\alpha }'; E,\varvec{\alpha }) f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha }') -s^{\mathrm {s}}_{-1}(E,\varvec{\alpha };E-\epsilon _p,\varvec{\alpha }') f^{\mathrm s}_h(t,E,\varvec{\alpha })]\chi \left( \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha })}{g(E)}\right) \nonumber \\&\qquad + \sum _{i=1}^{N_E}\int _{I_i(E+\epsilon _p)}C(E-\epsilon _p)\,\mathrm {d}\mu ^{E} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'}\nonumber \\&[s^{\mathrm {s}}_{-1}(E,\varvec{\alpha }';E-\epsilon _p,\varvec{\alpha }) f^{\mathrm s}_h(t,E,\varvec{\alpha }') -s^{\mathrm {s}}_{1}(E-\epsilon _p,\varvec{\alpha };E,\varvec{\alpha }') f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })]\chi \left( \frac{f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })}{g(E-\epsilon _p)}\right) \end{aligned}$$

(52)

[Exchange $\varvec{\alpha }$ and $\varvec{\alpha }'$ in the first summation above]

$$\begin{aligned}&= \sum _{i=1}^{N_E} \int _{I_i(E)}C(E-\epsilon _p)\,\mathrm {d}\mu ^{E} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'}\\&\quad [s^{\mathrm {s}}_{1}(E-\epsilon _p,\varvec{\alpha };E,\varvec{\alpha }') f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha }) -s^{\mathrm {s}}_{-1}(E,\varvec{\alpha }';E-\epsilon _p,\varvec{\alpha }) f^{\mathrm s}_h(t,E,\varvec{\alpha }')]\chi \left( \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha }')}{g(E)}\right) \end{aligned}$$

[In the second summation above, note that when $E\in [0,\epsilon _p] \cup [E_{\mathrm {max}},E_{\mathrm {max}}+\epsilon _p]$, $s_{-1}(E,\varvec{\alpha }';E-\epsilon _p,\varvec{\alpha }) =s_{1}(E-\epsilon _p,\varvec{\alpha };E,\varvec{\alpha }') =0$]

$$\begin{aligned}&+ \sum _{i=1}^{N_E}\int _{I_i(E)}C(E-\epsilon _p)\,\mathrm {d}\mu ^{E} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'}\\&\quad [s^{\mathrm {s}}_{-1}(E,\varvec{\alpha }';E-\epsilon _p,\varvec{\alpha }) f^{\mathrm s}_h(t,E,\varvec{\alpha }') -s^{\mathrm {s}}_{1}(E-\epsilon _p,\varvec{\alpha };E,\varvec{\alpha }') f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })]\chi \left( \frac{f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })}{g(E-\epsilon _p)}\right) . \end{aligned}$$

Using (27), we transform the above integration into

$$\begin{aligned}&\int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }}\sum _{i=1}^{N_E} \left( {\hat{S}}^{\mathrm {s}}\left[ f^{\mathrm s}_h(t,E,\varvec{\alpha }) \chi \left( \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha })}{g(E)}\right) \right] \right) _{i}\nonumber \\&\quad = -\sum _{i=1}^{N_E}\int _{I_i(E)}C(E-\epsilon _p)\,\mathrm {d}\mu ^{E} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{\varOmega }\,\mathrm {d}\mu ^{\varvec{\alpha }'} s^{\mathrm {s}}_{-1}(E,\varvec{\alpha }';E-\epsilon _p,\varvec{\alpha })g(E)\nonumber \\&\left[ \frac{f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })}{g(E-\epsilon _p)} - \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha }')}{g(E)}\right] \left[ \chi \left( \frac{f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })}{g(E-\epsilon _p)}\right) -\chi \left( \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha }')}{g(E)}\right) \right] \le 0, \end{aligned}$$

(53)

because $\chi$ is nondecreasing and the other part of the integrand is nonnegative. We let $\chi (x) = x$ in (53) and the resulted inequality becomes equality if and only if

$$\begin{aligned} \left[ \frac{f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })}{g(E-\epsilon _p)} - \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha }')}{g(E)}\right] ^2 = 0, \quad \text {for almost all} ~E,\varvec{\alpha },\varvec{\alpha }', \end{aligned}$$

thus

$$\begin{aligned} \frac{f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })}{g(E-\epsilon _p)}= \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha }')}{g(E)} \end{aligned}$$

holds almost everywhere. Based on the above discussion, we infer that there exists $\lambda > 0$ such that

$$\begin{aligned} \frac{f^{\mathrm s}_h(t,E-\epsilon _p,\varvec{\alpha })}{g(E-\epsilon _p)} = \frac{f^{\mathrm s}_h(t,E,\varvec{\alpha }')}{g(E)} = \lambda , \quad \text {for almost all} ~E,\varvec{\alpha },\varvec{\alpha }'. \end{aligned}$$

It is then concluded that

$$\begin{aligned} f^{\mathrm s}_h(t,E,\varvec{\alpha }) = \lambda g(E), \quad \text {for almost all } E \text { and } \varvec{\alpha }\end{aligned}$$

which proves the Lemma. $\square$

Proof of Theorem 3

We need to prove there exist $D_{k} > 0$, $k=1,\ldots ,m_i$, such that $M^{\mathrm {f}}_i {{\varvec{g}}}^{\mathrm {f}}_i = {\varvec{0}}$. Clearly, with the definition of ${{\varvec{g}}}_i^{\mathrm {f}}$ in (36), to solve $M_i^{\mathrm {f}}{{\varvec{g}}}_i^{\mathrm {f}} = {\varvec{0}}$ is equivalent to solve

$$\begin{aligned} H_i {\varvec{D}}_i = {\varvec{0}}, \quad {\varvec{D}}_i = \left[ \begin{array}{c} D_{i_1} \\ \vdots \\ D_{m_i} \end{array}\right] , \end{aligned}$$

(54)

in which

$$\begin{aligned} H_i = \lambda \varLambda L {\bar{H}}_i, \end{aligned}$$

(55)

where $\lambda = \int _{\varOmega } \,\mathrm {d}\mu ^{\varvec{\alpha }}$ and

$$\begin{aligned} \quad L = I_{m_i\times m_i}\otimes \left[ 1 , \ldots , 1 \right] _{1\times N_{\varvec{\alpha }}}^T,\quad \varLambda = I_{m_i\times m_i}\otimes \mathrm {diag}\left\{ \int _{S_1} \,\mathrm {d}\mu ^{\varvec{\alpha }},\ldots , \int _{S_{N_{\varvec{\alpha }}}} \,\mathrm {d}\mu ^{\varvec{\alpha }}\right\} , \end{aligned}$$

where $\otimes$ is the Kronecker product.

${\bar{H}}_i$ in (55) is a $n_i\hbox {th}$ order square matrix, and the $(a,b)\hbox {th}$ entry of ${\bar{H}}_i$ reads

$$\begin{aligned} {\bar{H}}_{i,ab}&= \int _{I_{\sigma _i^E(a)}(E-\epsilon _p)\cap I_{\sigma _i^E(b)}(E)}\,\mathrm {d}\mu ^{E}C(E+\epsilon _p) s_1^{\mathrm {npo}}(E,E+\epsilon _p) \\&\quad + \int _{I_{\sigma _i^E(a)}(E+\epsilon _p)\cap I_{\sigma _i^E(b)}(E)}\,\mathrm {d}\mu ^{E} C(E-\epsilon _p) s_{-1}^{\mathrm {npo}}(E,E-\epsilon _p) \\&\quad - \delta _{ab} \int _{I_{\sigma _i^E(a)}(E)}\,\mathrm {d}\mu ^{E} [s_{-1}^{\mathrm {npo}}(E,E-\epsilon _p)C(E-\epsilon _p) +s_{1}^{\mathrm {npo}}(E,E+\epsilon _p)C(E+\epsilon _p)]. \end{aligned}$$

Lemma 4 and a few calculations verify that ${\bar{H}}_i$ satisfies all the conditions listed in Table 4, thus $\mathrm {rank}({\bar{H}}_i) = n_i - 1$ and the existence of unique positive $(D_1,\ldots ,D_{m_i})^T \in \mathrm {Ker}({\bar{H}}_i)$ are obtained according to Lemma 5. Since $\varLambda$, L are both full column-rank matrices and

$$\begin{aligned} \mathrm {rank}(\varLambda ) + \mathrm {rank}(L) + \mathrm {rank}({\bar{H}}_i) -\mathrm {cl}(\varLambda ) - \mathrm {cl}(L) \le \mathrm {rank}(H_i) \le \mathrm {min}\{\mathrm {rank}(\varLambda ), \mathrm {rank}(L),\mathrm {rank}({\bar{H}}_i)\}, \end{aligned}$$

is set up, where $\mathrm {cl}(A)$ is the number of columns of matrix A, we obtain

$$\begin{aligned} \mathrm {rank}(H_i) = \mathrm {rank}({\bar{H}}_i) = m_i-1. \end{aligned}$$

(56)

It is then concluded from (56), (54) and (55) that

$$\begin{aligned} \mathrm {Ker}(H_i) = \mathrm {Ker}({\bar{H}}_i), \end{aligned}$$

(57)

and the theorem is established by recalling our former discussions about ${\bar{H}}_i$. $\square$

Proof of Theorem 4

Since $M^{\mathrm {f}}$ is irreducible and recalling Lemma 4, we could immediately obtain that the equilibrium ${{\varvec{g}}}^{\mathrm {f}}$ of (28) is unique up to multiplication by a constant, thus, the exsistance and uniqueness of $f^{\mathrm {eq}}_h(E,\varvec{\alpha })$ as the equilibrium of (23) is thus obtained.

Furthermore, since Platonic solids and regular pyramids are highly symmetrical, by using conditions (C1), (C2), and (13), we know each block of $M^{\mathrm {f}}$, saying $M^{\mathrm {f},jk}$, $1\le j,k\le N_E$, is a circulant matrix. In addition, the $(l,n)\hbox {th}$ entry of $M^{\mathrm {f},jk}$ reads

$$\begin{aligned}\begin{aligned} M^{\mathrm {f},jk}_{ln}&= \frac{1}{V_{kn}}\int _{I_j(E+\epsilon _p)\cap I_k(E)}\,\mathrm {d}\mu ^E\int _{S_l} \,\mathrm {d}\mu ^{\varvec{\alpha }}\int _{S_n} \,\mathrm {d}\mu ^{\varvec{\alpha }'} s_{-1}(E,\varvec{\alpha }'; E-\epsilon _p,\varvec{\alpha }) C(E-\epsilon _p) \\&= \frac{V_{jl}}{V_{kn}}\frac{1}{V_{jl}} \exp \left( \frac{\epsilon _p}{k_B T_L}\right) \int _{I_j(E)\cap I_k(E-\epsilon _p)}\,\mathrm {d}\mu ^E\int _{S_l} \,\mathrm {d}\mu ^{\varvec{\alpha }}\int _{S_n}\,\mathrm {d}\mu ^{\varvec{\alpha }'} s_{1}(E,\varvec{\alpha }'; E+\epsilon _p,\varvec{\alpha })C(E+\epsilon _p) \\&= \left( \exp \left( \frac{\epsilon _p}{k_B T_L}\right) \right) \frac{V_{jl}}{V_{kn}} M^{\mathrm {f,kj}}_{nl}, \quad \forall j < k. \end{aligned} \end{aligned}$$

Due to (C1) and (C2), $V_{jl}$ is constant with respect to l, saying

$$\begin{aligned} V_{jl} = D(j), \quad \forall ~ 1 \le j \le N_E, 1\le l\le N_{\varvec{\alpha }}, \end{aligned}$$

thus

$$\begin{aligned} M^{\mathrm {f},jk} = \exp \left( \frac{\epsilon _p}{k_BT_L}\right) \frac{D(j)}{D(k)} \left( M^{\mathrm {f},kj}\right) ^T, \quad \forall j < k. \end{aligned}$$

(58)

In all, by recalling $M^{\mathrm {f},jk}$ is circulant, $\forall 1\le j,k\le N_E$, the Zero row sum property of $M^{\mathrm {f}}$ as given in Table 4, and (58), one then concludes that there exists $(C_1,\ldots ,C_{N_E})^T$, where $C_i>0$, $i=1,\ldots ,N_E$, such that $f^{\mathrm {eq}}_h(E,\varvec{\alpha })$ has the form of (37). $\square$

C Properties of $M^{\mathrm {f}}$

By denoting

$$\begin{aligned}&s_{\tau _i(j),\tau _{i'}(j')}= \frac{1}{V_{i'j'}}\left[ \int _{I_i(E-\epsilon _p)\cap I_{i'}(E)}C(E+\epsilon _p)\,\mathrm {d}\mu ^E \int _{S_{j}}\,\mathrm {d}\mu ^{\varvec{\alpha }}\int _{S_{j'}} \,\mathrm {d}\mu ^{\varvec{\alpha }'} {\bar{s}}_1(E,\varvec{\alpha }';E+\epsilon _p,\varvec{\alpha }) \right. \nonumber \\&\quad \left. + \int _{I_i(E+\epsilon _p)\cap I_{i'}(E)}C(E-\epsilon _p)\,\mathrm {d}\mu ^E \int _{S_{j}}\,\mathrm {d}\mu ^{\varvec{\alpha }}\int _{S_{j'}} \,\mathrm {d}\mu ^{\varvec{\alpha }'} {\bar{s}}_{-1}(E,\varvec{\alpha }';E-\epsilon _p,\varvec{\alpha })\right] , \end{aligned}$$

(59)

and

$$\begin{aligned}&\lambda _{\tau _i(j),\tau _{i'}(j')} = \frac{\delta _{ii'}\delta _{jj'}}{V_{i'j'}} \int _{I_i(E)}\,\mathrm {d}\mu ^E\int _{S_{j}} \,\mathrm {d}\mu ^{\varvec{\alpha }}\int _{\varOmega } \,\mathrm {d}\mu ^{\varvec{\alpha }'} [{\bar{s}}_{-1}(E,\varvec{\alpha }; E-\epsilon _p,\varvec{\alpha }')C(E-\epsilon _p) \nonumber \\&\quad +\, {\bar{s}}_1(E,\varvec{\alpha }; E+\epsilon _p,\varvec{\alpha }')C(E+\epsilon _p)], \end{aligned}$$

(60)

we rewrite $M^{\mathrm {f}}_{\tau _i(j),\tau _{i'}(j)}$ in (29) as

$$\begin{aligned} M^{\mathrm {f}}_{\tau _i(j),\tau _{i'}(j)} = s_{\tau _i(j),\tau _{i'}(j)} - \delta _{ii'}\delta _{jj'} \lambda _{\tau _i(j),\tau _i(j)}. \end{aligned}$$

(61)

Noticing the definition of ${\bar{s}}_v(E,\varvec{\alpha };E',\varvec{\alpha }')$ given in (19) and (20), one sees

$$\begin{aligned} {\bar{s}}_v(E,\varvec{\alpha };E',\varvec{\alpha }') = {\bar{s}}_v(E,\varvec{\alpha }';E',\varvec{\alpha }), \quad \forall \varvec{\alpha },\varvec{\alpha }' \in \varOmega . \end{aligned}$$

(62)

Exchanging $\varvec{\alpha }$ and $\varvec{\alpha }'$ in the integrals of (59) and taking summation of all the entries of $M^{\mathrm {f}}$ in arbitrary column, we obtain

$$\begin{aligned} \sum _{ij} M^{\mathrm {f}}_{\tau _i(j),\tau _{i'}(j')} =\sum _{ij} s_{\tau _i(j),\tau _{i'}(j')} - \lambda _{\tau _{i'}(j'),\tau _{i'}(j')} = 0, \quad \forall 1 \le i'\le N_E, 1\le j'\in N_{\varvec{\alpha }}. \end{aligned}$$

(63)

Since ${\bar{s}}_v(E,\varvec{\alpha };E',\varvec{\alpha }') > 0$ in (59) and (60),

$$\begin{aligned} \begin{array}{cc} s_{\tau _i(j),\tau _{i'}(j')}\ge 0,&\lambda _{\tau _i(j),\tau _i(j)} > 0, \quad \forall 1\le i,i'\le N_E, 1\le j\le N_{\varvec{\alpha }}, 1\le j'\le N_{\varvec{\alpha }}, \end{array} \end{aligned}$$

(64)

is obvious. Considering (61), (63) and (64) together, we obtain that

$$\begin{aligned} M^{\mathrm {f}}_{\tau _i(j),\tau _{i'}(j')} \left\{ \begin{array}{ll} < 0, &{} \quad \text {if} \,\,i = i', j=j',\\ \ge 0, &{} \quad \text {otherwise}. \end{array}\right. \end{aligned}$$

(65)

Furthermore, one sees for arbitrary $1 \le i,i'\le N_E$ that

$$\begin{aligned} \exists 1\le j_0,j_0'\le N_{\varvec{\alpha }}, \quad s.t.\quad M^{\mathrm {f}}_{\tau _i(j_0),\tau _{i'}(j_0')} \ne 0 \Rightarrow \quad M^{\mathrm {f}}_{\tau _i(j),\tau _{i'}(j')} \ne 0, \quad \forall 1\le j,j'\le N_{\varvec{\alpha }}. \end{aligned}$$

(66)

According to (9) and (27), we have that

$$\begin{aligned}&V_{i'j'}s_{\tau _i(j),\tau _{i'}(j')} =\int _{I_i(E-\epsilon _p)\cap I_{i'}(E)}\,\mathrm {d}\mu ^E \int _{S_{j}} \,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{S_{j'}} \,\mathrm {d}\mu ^{\varvec{\alpha }'}{\bar{s}}_1(E,\varvec{\alpha }'; E+\epsilon _p,\varvec{\alpha }) C(E+\epsilon _p) \nonumber \\&\qquad + \int _{I_i(E+\epsilon _p)\cap I_{i'}(E)}\,\mathrm {d}\mu ^E \int _{S_{j}}\,\mathrm {d}\mu ^{\varvec{\alpha }} \int _{S_{j'}} \,\mathrm {d}\mu ^{\varvec{\alpha }'}{\bar{s}}_{-1}(E,\varvec{\alpha }'; E-\epsilon _p,\varvec{\alpha })C(E-\epsilon _p) \end{aligned}$$

(67)

$$\begin{aligned}&\quad = \exp \left( -\frac{\epsilon _p}{k_BT}\right) \int _{I_{i'}(E+\epsilon _p)\cap I_i(E)}\,\mathrm {d}\mu ^E \int _{S_{j}}\,\mathrm {d}\mu ^{\varvec{\alpha }}\int _{S_{j'}} \,\mathrm {d}\mu ^{\varvec{\alpha }'}{\bar{s}}_{-1}(E,\varvec{\alpha }'; E-\epsilon _p,\varvec{\alpha }) C(E-\epsilon _p) \nonumber \\&\qquad + \exp \left( \frac{\epsilon _p}{k_BT}\right) \int _{I_{i'}(E-\epsilon _p) \cap I_i(E)}\,\mathrm {d}\mu ^E \int _{S_{j}}\,\mathrm {d}\mu ^{\varvec{\alpha }}\int _{S_{j'}} \,\mathrm {d}\mu ^{\varvec{\alpha }'}{\bar{s}}_{1}(E,\varvec{\alpha }'; E+\epsilon _p,\varvec{\alpha }) C(E+\epsilon _p) \nonumber \\&\quad \in [C_\star ,C^\star ]\left( V_{ij}s_{\tau _{i'}(j'),\tau _i(j)}\right) , \end{aligned}$$

(68)

where

$$\begin{aligned}\begin{array}{cc} C_\star = \min \{\exp \left( -\frac{\epsilon _p}{k_BT}\right) , \exp \left( \frac{\epsilon _p}{k_BT}\right) \}> 0,&C^\star = \max \{\exp \left( -\frac{\epsilon _p}{k_BT}\right) , \exp \left( \frac{\epsilon _p}{k_BT}\right) \} > 0, \end{array}\end{aligned}$$

thus

$$\begin{aligned} M^{\mathrm {f}}_{\tau _i(j),\tau _{i'}(j')}> 0\Leftrightarrow M^{\mathrm {f}}_{\tau _{i'}(j'),\tau _i(j)} > 0,\quad \forall 1 \le i,i'\le N_E, 1\le j,j'\le N_{\varvec{\alpha }}. \end{aligned}$$

(69)