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Decay of Information for the Kac Evolution

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Abstract

We consider a system of M particles in contact with a heat reservoir of \(N\gg M\) particles. The evolution in the system and the reservoir, together with their interaction, is modeled via the Kac’s master equation. We chose the initial distribution with total energy \(N+M\) and show that if the reservoir is initially in equilibrium, that is, if the initial distribution depends only on the energy of the particle in the reservoir, then the entropy of the system decays exponentially to a very small value. We base our proof on a similar property for the Information. A similar argument allows us to greatly simplify the proof of the main result in Bonetto et al. (Commun Math Phys 363(3):847–875, 2018).

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References

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Correspondence to F. Bonetto.

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Communicated by Christian Maes.

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© 2017 by the authors. Reproduction of this article by any means permitted for non-commercial purposes. U.S. National Science Foundation Grants DMS-1907643 (F.B), DMS-2053285 (R.H.) and DMS-1856645 (M.L) are gratefully acknowledged.

Appendices

Appendix A: Proofs of Lemmas 4.2 and 4.3

1.1 A.1 Proof of Lemma 4.2

We have

$$\begin{aligned} \mathcal {I}: =&\int _{{\mathop {\mathbb {S}}}^{M+N-1}(\sqrt{M+N})} \frac{|L^{\mathrm{out}} (F\circ R^{-1})|^2}{F\circ R^{-1}} {\mathord {\mathrm {d}}}\sigma \\ =&\int _{{\mathop {\mathbb {S}}}^{M+N-1}(\sqrt{M+N})} \frac{\sum ^{{\mathrm {out}}} |\omega _i( C\nabla _{{\mathord {\mathbf {v}}}}F+D\nabla _{{\mathord {\mathbf {w}}}}F)_j-\omega _j ( SC\nabla _{{\mathord {\mathbf {v}}}}F+D\nabla _{{\mathord {\mathbf {w}}}}F)_i|^2}{F} {\mathord {\mathrm {d}}}\sigma , \end{aligned}$$

where the argument of the functions is of the form \((A^T{\mathord {\mathbf {v}}}+C^T {\mathord {\mathbf {w}}}, B^T{\mathord {\mathbf {v}}}+D^T {\mathord {\mathbf {w}}})\), and \(\sum ^{\mathrm{out}}:=\sum _{M+1\le i<j\le N+M}\). Next, we change variable \({\mathord {\mathbf {v}}}=A{\mathord {\mathbf {v}}}+B{\mathord {\mathbf {w}}}\), \({\mathord {\mathbf {w}}}=C{\mathord {\mathbf {v}}}+D{\mathord {\mathbf {w}}}\), we have

$$\begin{aligned} \mathcal {I}&=\int _{{\mathop {\mathbb {S}}}^{M+N-1}(\sqrt{M+N})} {\mathord {\mathrm {d}}}\sigma \\&\quad \frac{\sum ^{\mathrm {out}} |(C{\mathord {\mathbf {v}}}+D{\mathord {\mathbf {w}}})_i( C\nabla _{{\mathord {\mathbf {v}}}}F+D\nabla _{{\mathord {\mathbf {w}}}}F)_j-(C{\mathord {\mathbf {v}}}+D{\mathord {\mathbf {w}}})_j (C\nabla _{{\mathord {\mathbf {v}}}}F+D\nabla _{{\mathord {\mathbf {w}}}}F)_i|^2}{F}\\&=: \int _{{\mathop {\mathbb {S}}}^{M+N-1}(\sqrt{M+N})} \frac{\Sigma _0}{F} {\mathord {\mathrm {d}}}\sigma \end{aligned}$$

By (28), we have

$$\begin{aligned} \Sigma _0 =&\sum ^{\mathrm {out}} |(C{\mathord {\mathbf {v}}}+D{\mathord {\mathbf {w}}})_i( C\nabla _{{\mathord {\mathbf {v}}}}F+\ell D {\mathord {\mathbf {w}}})_j-(C{\mathord {\mathbf {v}}}+D{\mathord {\mathbf {w}}})_j (C\nabla _{{\mathord {\mathbf {v}}}}F+\ell D{\mathord {\mathbf {w}}})_i|^2\\ =&\sum ^{\mathrm {out}} |(C{\mathord {\mathbf {v}}}+D{\mathord {\mathbf {w}}})_i (C (\nabla _{{\mathord {\mathbf {v}}}} F-\ell {\mathord {\mathbf {v}}}))_j-(C{\mathord {\mathbf {v}}}+D{\mathord {\mathbf {w}}})_j (C (\nabla _{{\mathord {\mathbf {v}}}} F-\ell {\mathord {\mathbf {v}}}))_i|^2. \end{aligned}$$

Now we average over the directions of the vector \({\mathord {\mathbf {w}}}\). Recall that the function F does not depend on this direction nor does \(\ell \). Denote such an average by \(\langle \cdot \rangle \). Hence, we get, after averaging over each pair (ij),

$$\begin{aligned} \langle \Sigma _0\rangle&=\sum ^{\mathrm {out}} |(C{\mathord {\mathbf {v}}})_i(C (\nabla _{{\mathord {\mathbf {v}}}} F-\ell {\mathord {\mathbf {v}}}))_j-(C{\mathord {\mathbf {v}}})_j C (\nabla _{{\mathord {\mathbf {v}}}} F-\ell {\mathord {\mathbf {v}}}))_i|^2\\&\quad +\sum ^{\mathrm {out}}\Big \langle |(D{\mathord {\mathbf {w}}})_i(C (\nabla _{{\mathord {\mathbf {v}}}} F-\ell {\mathord {\mathbf {v}}}))_j-(D{\mathord {\mathbf {w}}})_j(C (\nabla _{{\mathord {\mathbf {v}}}} F-\ell {\mathord {\mathbf {v}}}))_i|^2 \Big \rangle \\&= |C{\mathord {\mathbf {v}}}|^2 |C \nabla _{{\mathord {\mathbf {v}}}} F |^2-(C{\mathord {\mathbf {v}}}\cdot C \nabla _{{\mathord {\mathbf {v}}}} F)^2\\&\quad +\Big \langle |D{\mathord {\mathbf {w}}}|^2 \Big \rangle |C (\nabla _{{\mathord {\mathbf {v}}}} F-\ell {\mathord {\mathbf {v}}})|^2-\Big \langle ({\mathord {\mathbf {w}}}, D^T C (\nabla _{{\mathord {\mathbf {v}}}} F-\ell {\mathord {\mathbf {v}}}))^2\Big \rangle \end{aligned}$$

where we used that \(\langle \omega _i\rangle =0\). Next, by using that

$$\begin{aligned} \langle \omega _i\, \omega _j\rangle =\frac{|{\mathord {\mathbf {w}}}|^2}{N} \delta _{ij}, \end{aligned}$$

we have

$$\begin{aligned} \langle \Sigma _0\rangle =&|C {\mathord {\mathbf {v}}}|^2 |C \nabla _{\mathord {\mathbf {v}}}F|^2 -(C {\mathord {\mathbf {v}}}\cdot C\nabla _{\mathord {\mathbf {v}}}F)^2 \\&+\frac{|{\mathord {\mathbf {w}}}|^2}{N} \left[ {\mathrm{Tr }} (D^T D) |C(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2 -|D^TC(\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F)|^2 \right] \end{aligned}$$

Finally, just note that due to the fact that F does not depend on the direction of \({\mathord {\mathbf {w}}}\), we have

$$\begin{aligned} \mathcal {I}= \int _{{\mathop {\mathbb {S}}}^{M+N-1}(\sqrt{M+N})} \frac{\langle \Sigma _0 \rangle }{F} {\mathord {\mathrm {d}}}\sigma . \end{aligned}$$

This proves the lemma. \(\square \)

1.2 A.2 Proof of Lemma 4.3

Combining Lemma 4.1 with 4.2, it suffices to show

$$\begin{aligned} |LF|^2-\Sigma _1\le \Sigma _2. \end{aligned}$$
(30)

Using the rotation invariance of F in the second variable, we write

$$\begin{aligned} |LF|^2= & {} \sum _{i<j} |v_i\partial _{v_j} F - v_j \partial _{v_i} F|^2 + \sum _i \sum _j |w_j \partial _{v_i}F - v_i \partial _{w_j} F|^2. \\= & {} |{\mathord {\mathbf {v}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 - ({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F)^2 + |{\mathord {\mathbf {w}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 + |{\mathord {\mathbf {v}}}|^2 |{\mathord {\mathbf {w}}}|^2 \ell ^2 - 2|{\mathord {\mathbf {w}}}|^2 \ell ({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F) \\= & {} |{\mathord {\mathbf {v}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 - ({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F)^2 + |{\mathord {\mathbf {w}}}|^2 |\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F|^2 . \end{aligned}$$

It remains, thus, to estimate

$$\begin{aligned}&|{\mathord {\mathbf {v}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 - ({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F)^2 + |{\mathord {\mathbf {w}}}|^2 |\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F|^2 \nonumber \\&\quad - |C {\mathord {\mathbf {v}}}|^2 |C \nabla _{\mathord {\mathbf {v}}}F|^2 + (C {\mathord {\mathbf {v}}}\cdot C\nabla _{\mathord {\mathbf {v}}}F)^2 \nonumber \\&\quad -\frac{|{\mathord {\mathbf {w}}}|^2}{N} \left[ |C(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2{\mathrm{Tr }} D^TD -|D^TC(\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F)|^2 \right] \end{aligned}$$
(31)

Since R is a rotation, we have that

$$\begin{aligned} \left( \begin{array}{cc} A &{} B \\ C &{} D \end{array} \right) \left( \begin{array}{cc} A^T &{} C^T \\ B^T &{} D^T \end{array} \right) = \left( \begin{array}{cc} I_M &{} 0 \\ 0 &{} I_N \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \left( \begin{array}{cc} A^T &{} C^T \\ B^T &{} D^T \end{array} \right) \left( \begin{array}{cc} A &{} B \\ C &{} D \end{array} \right) =\left( \begin{array}{cc} I_M &{} 0 \\ 0 &{} I_N \end{array} \right) \ . \end{aligned}$$

Using this we get that

$$\begin{aligned} {\mathrm{Tr}} D^TD = N-{\mathrm{Tr}}B^TB = N-M + {\mathrm{Tr}}A^TA , A^TA+C^TC = I_M \end{aligned}$$

and

$$\begin{aligned} D^TC = -B^TA . \end{aligned}$$

Using these relations (31) becomes

$$\begin{aligned}&|{\mathord {\mathbf {v}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 - ({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F)^2 + |{\mathord {\mathbf {w}}}|^2 |\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F|^2 \\&\quad - (| {\mathord {\mathbf {v}}}|^2 - |A {\mathord {\mathbf {v}}}|^2)( | \nabla _{\mathord {\mathbf {v}}}F|^2-|A \nabla _{\mathord {\mathbf {v}}}F|^2) + \left[ ({\mathord {\mathbf {v}}}, \nabla _{\mathord {\mathbf {v}}}F) -( A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F)\right] ^2 \\&\quad -\frac{|{\mathord {\mathbf {w}}}|^2}{N} \left[ ( |(\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F)|^2-|A(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2)(N-{\mathrm{Tr }}B^TB) -|B^TA(\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F)|^2 \right] \end{aligned}$$

which can be simplified to

$$\begin{aligned}&|{\mathord {\mathbf {v}}}|^2 |A\nabla _{\mathord {\mathbf {v}}}F|^2 +|A{\mathord {\mathbf {v}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 - |A{\mathord {\mathbf {v}}}|^2 |A\nabla _{\mathord {\mathbf {v}}}F|^2 + (A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F)^2 \\&\quad -2({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F) (A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F) \\&\quad +|{\mathord {\mathbf {w}}}|^2 |A(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2 \\&\quad +\frac{|{\mathord {\mathbf {w}}}|^2}{N} \left[ ( |(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2-|A(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2){\mathrm{Tr }}B^TB +|B^TA(\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F)|^2 \right] . \end{aligned}$$

Since \( AA^T+BB^T = I_M\) and \({\mathrm{Tr}} B^TB = {\mathrm{Tr}} BB^T = M - {\mathrm{Tr}} AA^T\), we obtain an upper bound

$$\begin{aligned}&|{\mathord {\mathbf {v}}}|^2 |A\nabla _{\mathord {\mathbf {v}}}F|^2 +|A{\mathord {\mathbf {v}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 - |A{\mathord {\mathbf {v}}}|^2 |A\nabla _{\mathord {\mathbf {v}}}F|^2 + (A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F)^2\\&\quad -2({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F) (A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F) \\&\quad +|{\mathord {\mathbf {w}}}|^2 |A(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2 \\&\quad +\frac{|{\mathord {\mathbf {w}}}|^2}{N} \left[ ( |(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2-|A(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2)M +|A(\ell {\mathord {\mathbf {v}}}- \nabla _{\mathord {\mathbf {v}}}F)|^2 \right] . \end{aligned}$$

which can be simplified to

$$\begin{aligned}&|{\mathord {\mathbf {v}}}|^2 |A\nabla _{\mathord {\mathbf {v}}}F|^2 +|A{\mathord {\mathbf {v}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 - |A{\mathord {\mathbf {v}}}|^2 |A\nabla _{\mathord {\mathbf {v}}}F|^2 + (A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F)^2 \\&\quad -2({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F) (A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F) \\&\quad + \left( 1-\frac{M-1}{N}\right) |{\mathord {\mathbf {w}}}|^2 |A(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2 + \frac{M}{N} |{\mathord {\mathbf {w}}}|^2 |(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2 . \end{aligned}$$

Regarding the terms quartic in A we do not know how to handle but noting that \(- |A{\mathord {\mathbf {v}}}|^2 |A\nabla _{\mathord {\mathbf {v}}}F|^2 + (A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F)^2\le 0\) we get the bound

$$\begin{aligned}&|LF|^2-\Sigma _1\le |{\mathord {\mathbf {v}}}|^2 |A\nabla _{\mathord {\mathbf {v}}}F|^2 +|A{\mathord {\mathbf {v}}}|^2 |\nabla _{\mathord {\mathbf {v}}}F|^2 - 2({\mathord {\mathbf {v}}}\cdot \nabla _{\mathord {\mathbf {v}}}F) (A {\mathord {\mathbf {v}}}\cdot A\nabla _{\mathord {\mathbf {v}}}F) \\&\quad + \left( 1-\frac{M-1}{N}\right) |{\mathord {\mathbf {w}}}|^2 |A(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2 + \frac{M}{N} |{\mathord {\mathbf {w}}}|^2 |(\ell {\mathord {\mathbf {v}}}-\nabla _{\mathord {\mathbf {v}}}F)|^2=\Sigma _2 , \end{aligned}$$

as claimed. \(\square \)

A.3 Proof of Lemma 2.1

One can think about K as the top left entry of the matrix

$$\begin{aligned} \sum _{\alpha _1, \dots , \alpha _k} \lambda _{\alpha _1} \cdots \lambda _{\alpha _k} \int _{[-\pi ,\pi ]^k} \nu (\mathrm {d} \theta _1) \cdots \nu (\mathrm {d} \theta _k) \, \left[ \prod _{l=1}^k r_{\alpha _l}(\theta _l)\right] ^{-1} \begin{pmatrix} I_M &{} 0 \\ 0 &{} 0\end{pmatrix} \left[ \prod _{l=1}^k r_{\alpha _l}(\theta _l)\right] . \end{aligned}$$

The computation hinges on a repeated application of the elementary identity

$$\begin{aligned}&\int _{-\pi }^{\pi } \nu (\mathrm {d}\theta ) \, \begin{pmatrix}\cos (\theta ) &{} -\sin (\theta )\\ \sin (\theta ) &{} \cos (\theta )\end{pmatrix} \begin{pmatrix} m_1 &{} 0\\ 0 &{} m_2\end{pmatrix} \begin{pmatrix} \cos (\theta ) &{} \sin (\theta )\\ -\sin (\theta ) &{} \cos (\theta )\end{pmatrix} \\&\quad = \begin{pmatrix} (1-\tilde{\nu }) m_1+\tilde{\nu } m_2 &{} 0\\ 0 &{} (1-\tilde{\nu }) m_2+\tilde{\nu } m_1 \end{pmatrix} , \end{aligned}$$

where \(\tilde{\nu }=\int \nu (\mathrm {d} \theta ) \, \sin ^2(\theta ).\) For this to be true, we just need (9). We easily check that for the rotations \(r_\alpha (\theta )\)

$$\begin{aligned}&\sum _{\alpha } \lambda _{\alpha } \int _{-\pi }^{\pi } \nu (\mathrm {d} \theta ) \, r_\alpha (\theta )^{-1} \begin{pmatrix} m_1 I_M &{} 0 \\ 0 &{}m_2 I_N\end{pmatrix} r_\alpha (\theta )\nonumber \\&\quad = \frac{1}{\Lambda }\left( \frac{M\lambda _S}{2}+\frac{N\lambda _R}{2}\right) \begin{pmatrix} m_1 I_M &{} 0 \\ 0 &{} m_2 I_N\end{pmatrix} \nonumber \\&\qquad + \frac{\mu }{\Lambda N}\begin{pmatrix} N(M-1)+N((1-\tilde{\nu }) m_1 + \tilde{\nu } m_2)I_M &{} 0 \\ 0 &{} (N-1)M+ M(\tilde{\nu } m_1 +(1-\tilde{\nu })m_2) I_N \end{pmatrix} \nonumber \\&\quad = \begin{pmatrix} m_1 I_M &{} 0 \\ 0 &{} m_2 I_N\end{pmatrix}+ \frac{\mu _\nu }{\Lambda N} \begin{pmatrix} N( m_2-m_1)I_M &{} 0 \\ 0 &{} M(m_1 -m_2)I_N \end{pmatrix} . \end{aligned}$$
(32)

where \(\mu _\nu =\tilde{\nu } \mu \). Denote by \(L(\nu _1,\nu _2)\) the \((N+M)\times (N+M)\) matrix

$$\begin{aligned} L(m_1,m_2)=\begin{pmatrix} m_1 I_{M} &{} 0\\ 0 &{} m_2 I_N\end{pmatrix}, \end{aligned}$$

and set

$$\begin{aligned} \mathcal {P}=I_2 - \frac{\mu _\nu }{\Lambda N}\begin{pmatrix} N &{} -N\\ -M &{} M \end{pmatrix} . \end{aligned}$$

Then, (32) is recast as

$$\begin{aligned} \sum _{\alpha } \lambda _{\alpha } \int _{-\pi }^{\pi } \nu (\mathrm {d} \theta ) \, r_\alpha (\theta )^{-1}L(m_1,m_2)r_\alpha (\theta )= L(m_1',m_2') , \end{aligned}$$
(33)

where

$$\begin{aligned} \begin{pmatrix} m_1' \\ m_2'\end{pmatrix}= \mathcal {P}\begin{pmatrix} m_1 \\ m_2\end{pmatrix} . \end{aligned}$$

By a repeated application of (33), we obtain

$$\begin{aligned}&\sum _{\alpha _1, \dots , \alpha _k} \lambda _{\alpha _1} \cdots \lambda _{\alpha _k} \int _{[-\pi ,\pi ]^k} \nu (\mathrm {d}\theta _1) \, \cdots \nu (\mathrm {d}\theta _k) \, \left[ \prod _{j=1}^k r_{\alpha _j}(\theta _j)\right] ^T L(\underline{m})\left[ \prod _{j=1}^k r_{\alpha _j}(\theta _j)\right] \\&\quad = L(\mathcal {P}^k\underline{m}) . \end{aligned}$$

Thus,

$$\begin{aligned} K =\left( \mathcal {P}^k\, \begin{pmatrix} 1 \\ 0\end{pmatrix}\right) _1 I_M . \end{aligned}$$

It is easy to see that \(\mathcal {P}\) has eigenvalues \(\ell _1=1\) and \(\ell _2=1-\mu _\nu (M+N)/(\Lambda N)\) with eigenvectors \(\underline{m}_1=(1, 1)\) and \(\underline{m}_2=(N,-M)^T/(M+N)\). Consequently,

$$\begin{aligned} \begin{pmatrix} 1 \\ 0\end{pmatrix}=\frac{M}{N+M}\underline{m}_1+\underline{m}_2 , \end{aligned}$$

which yields

$$\begin{aligned} \left( \mathcal {P}^k\, \begin{pmatrix} 1 \\ 0\end{pmatrix}\right) _1=\frac{M}{N+M}+ \frac{N}{M+N}\left( 1-\mu _\nu \frac{M+N}{\Lambda N}\right) ^k . \end{aligned}$$

This proves Lemma 2.1. \(\square \)

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Bonetto, F., Han, R. & Loss, M. Decay of Information for the Kac Evolution. Ann. Henri Poincaré 22, 2975–2993 (2021). https://doi.org/10.1007/s00023-021-01050-3

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