Proof of The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality

Each result cited is universally quantified over the data in its own statement.

Throughout, $\Pi_{N}$ is as in The Logarithmic Energy of N Ordered Particles on the Weyl Chamber, so that $H(x)=-\beta\sum_{(i,j)\in\Pi_{N}}\log(x_{i}-x_{j})$; $\mathbb{R}$ is identified with $\mathbb{R}^{1}$ as in the statement; for $x\in\mathbb{R}^{N}$ we write $A(x)=\sum_{k=1}^{N}|x_{k}|$; and for $x\in W_{N}$ we write $v(x)\in\mathbb{R}^{N}$ for the point with $k$th coordinate $v_{k}(x)=V_{1}'(x_{k})$. Finite sums of real numbers are compared termwise: if $a_{k}\le b_{k}$ for every $k\in[n]$ then $\sum_{k=1}^{n}a_{k}\le\sum_{k=1}^{n}b_{k}$, by claims 2, 3 and 5 of Properties of Finite Sums applied to the differences $b_{k}-a_{k}$ together with claim 3 of Elementary Arithmetic in an Ordered Field; and $\sum_{k=1}^{n}c=nc$ for a constant $c$, by induction on $n$ from the recursion in claim 1 there. Elementary order arithmetic is from Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field.

Step 0 (the function $V_{1}'$). By clause 2 of C^k Maps on a Euclidean Open Set (with $k=1$), $V_{1}$ is of class $C^{1}$ on $\mathbb{R}$ and $V_{1}'=\partial_{1}V_{1}$ is of class $C^{1}$ on $\mathbb{R}$; by clauses 1 and 4 there, the partial derivative of $V_{1}'$ with respect to the first variable exists at every $t\in\mathbb{R}$ with value $V_{1}''(t)$, so by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line the function $V_{1}'$ is differentiable at every $t\in\mathbb{R}$ with $(V_{1}')'(t)=V_{1}''(t)$. By claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous (with $U=\mathbb{R}$, $k=1$), $V_{1}'$ is continuous at every point of $\mathbb{R}$ relative to $\mathbb{R}$ as a map into $(\mathbb{R},d_{\mathbb{R}})$, that is, continuous on $\mathbb{R}$. By claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, $\mathbb{R}$ is an interval every point of which is interior, and it contains the two points $0$ and $1$.

(0a) Let $a,b\in\mathbb{R}$ with $a<b$. The restriction $g$ of $V_{1}'$ to $[a,b]$ is continuous on $[a,b]$ by claim 1 of Restriction Stability of Continuity and of the Derivative; $[a,b]$ is an interval of which every $c$ with $a<c<b$ is an interior point (The Real Line: Standing Notation and Background for Calculus §intervals), so by claim 2 of Restriction Stability of Continuity and of the Derivative $g$ is differentiable at such $c$ with $g'(c)=V_{1}''(c)$. By Mean Value Theorem on a Closed Real Interval there is $c$ with $a<c<b$ and

(0b) If $0\le V_{1}''(t)$ for every $t\in\mathbb{R}$, then by The Sign of the Derivative and Monotonicity §nondecreasing (with $I=\mathbb{R}$ and $f=V_{1}'$) the function $V_{1}'$ is nondecreasing on $\mathbb{R}$: $V_{1}'(s)\le V_{1}'(t)$ whenever $s\le t$.

Claim 1. Fix $k\in[N]$ and let $\pi_{k}:\mathbb{R}^{N}\to\mathbb{R}$, $\pi_{k}(x)=x_{k}$. The set $\mathbb{R}^{N}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and $\pi_{k}$ is smooth on it by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set and of class $C^{1}$ by claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. For $x\in\mathbb{R}^{N}$, $i\in[N]$ and real $h\ne0$, the point obtained from $x$ by replacing $x_{i}$ with $x_{i}+h$ lies in $\mathbb{R}^{N}$, and the difference quotient of $\pi_{k}$ there is exactly $(x_{k}+h-x_{k})h^{-1}=1$ if $i=k$ and $0\cdot h^{-1}=0$ if $i\ne k$; so by Partial Derivative on a Euclidean Open Set, $\partial_{i}\pi_{k}(x)$ is $1$ if $i=k$ and $0$ otherwise. Since $\pi_{k}$ takes values in the open set $\mathbb{R}$ and $V_{1}$ is of class $C^{2}$ on $\mathbb{R}$, claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ shows that $\varphi_{k}=V_{1}\circ\pi_{k}$, $\varphi_{k}(x)=V_{1}(x_{k})$, is of class $C^{2}$ on $\mathbb{R}^{N}$, and claim 1 there (with $m=p=1$, the sum over $l$ having the single term $l=1$ by claim 1 of Properties of Finite Sums) gives $\partial_{i}\varphi_{k}(x)=V_{1}'(x_{k})\,\partial_{i}\pi_{k}(x)$, which is $V_{1}'(x_{k})$ if $i=k$ and $0$ otherwise.

Let $\Phi=\sum_{k=1}^{N}\varphi_{k}$ on $\mathbb{R}^{N}$. By induction on the number of summands, claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set shows that $\Phi$ is of class $C^{2}$ on $\mathbb{R}^{N}$, and claim 1 there together with claim 7 of Properties of Finite Sums gives $\partial_{i}\Phi(x)=\sum_{k=1}^{N}\partial_{i}\varphi_{k}(x)=V_{1}'(x_{i})$ for $x\in\mathbb{R}^{N}$ and $i\in[N]$. Thus $\partial_{i}\Phi=V_{1}'\circ\pi_{i}$; as $V_{1}'$ is of class $C^{1}$ on $\mathbb{R}$ (Step 0) and $\pi_{i}$ is of class $C^{1}$, claim 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ gives

The set $W_{N}\subseteq\mathbb{R}^{N}$ is open by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open, and $P=H+\Phi|_{W_{N}}$. By Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives (resting on claims 1 and 3 of Restriction of a $C^k$ Map to an Open Subset), $\Phi|_{W_{N}}$ is of class $C^{2}$ on $W_{N}$ and its first and second partial derivatives at points of $W_{N}$ are those of $\Phi$. Since $H$ is of class $C^{2}$ on $W_{N}$ by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §derivatives, claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set shows that $P$ is of class $C^{2}$ on $W_{N}$, and claim 1 there gives, for $x\in W_{N}$ and $k\in[N]$,

Applying claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set once more to $\partial_{k}P=\partial_{k}H+\partial_{k}(\Phi|_{W_{N}})$, whose two summands have partial derivatives at every point of $W_{N}$ because $H$ and $\Phi|_{W_{N}}$ are of class $C^{2}$ (clauses 1, 2 and 4 of C^k Maps on a Euclidean Open Set), gives $\partial_{k}\partial_{k}P(x)=\partial_{k}\partial_{k}H(x)+V_{1}''(x_{k})$. By Hessian Matrix of a C^2 Function and Trace of a Real Square Matrix, claim 2 of Properties of Finite Sums and The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §derivatives,

In particular, by Gradient of a Real-Valued Function on a Euclidean Open Set, the $k$th coordinate of $DP(x)$ is $\partial_{k}H(x)+v_{k}(x)$.

Claim 2. $W_{N}$ is open and $P$ is of class $C^{2}$ on $W_{N}$ by claim 1, so clause 1 of Penalty on an Open Subset of Euclidean Space holds, and it remains to show that $K_{t}=\{x\in W_{N}:P(x)\le t\}$ is compact for every $t\in\mathbb{R}$.

(2a) Let $\mu>0$, $x\in W_{N}$ and $(i,j)\in\Pi_{N}$, and put $s=x_{i}-x_{j}$, which is positive. Since $s=(s\mu^{-1})\mu$ with $s\mu^{-1}>0$ (claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field), The Natural Logarithm gives $\log s=\log(s\mu^{-1})+\log\mu$, and The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log gives $\log(s\mu^{-1})\le s\mu^{-1}-1$. By claim 3 of Properties of the Absolute Value in an Ordered Field, $x_{i}\le|x_{i}|$ and $-x_{j}\le|x_{j}|$, so $s\le|x_{i}|+|x_{j}|$; and $|x_{i}|,|x_{j}|\le A(x)$ by claim 6 of Properties of Finite Sums, the summands $|x_{k}|$ being nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field. Multiplying by $-\beta<0$ (claim 4 of Elementary Order Arithmetic in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field) we obtain, with $T_{ij}(x)=-\beta\log(x_{i}-x_{j})$,

(2b) Let $n_{\Pi}=\sum_{p\in\Pi_{N}}1$. Since $(1,2)\in\Pi_{N}$, claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set and Nonnegativity and Monotonicity of a Sum over a Finite Index Set §monotone give $1\le n_{\Pi}$, so $0<n_{\Pi}$. By claim 4 of Properties of a Sum over a Finite Index Set, $H(x)=\sum_{(i,j)\in\Pi_{N}}T_{ij}(x)$, so (2.1), Nonnegativity and Monotonicity of a Sum over a Finite Index Set §comparison and claims 3 and 4 of Properties of a Sum over a Finite Index Set give, for every $\mu>0$ and $x\in W_{N}$,

Take $\mu_{0}=4\beta n_{\Pi}a_{0}^{-1}>0$, so that $2\beta\mu_{0}^{-1}n_{\Pi}=\tfrac{a_{0}}{2}$, and put $c_{0}=n_{\Pi}\beta(1-\log\mu_{0})$; then $H(x)\ge-\tfrac{a_{0}}{2}A(x)+c_{0}$. Summing the hypothesis $V_{1}(x_{k})\ge a_{0}|x_{k}|-b_{0}$ over $k$ gives $\sum_{k=1}^{N}V_{1}(x_{k})\ge a_{0}A(x)-Nb_{0}$. With $C_{1}=Nb_{0}-c_{0}$ and claim 8 of Elementary Order Arithmetic in an Ordered Field,

(2c) Fix $t\in\mathbb{R}$ and let $R_{t}=\max\{2a_{0}^{-1}(t+C_{1}),1\}>0$. For $x\in K_{t}$, (2.3) gives $\tfrac{a_{0}}{2}A(x)\le t+C_{1}$, so $A(x)\le R_{t}$ by claim 1 of Elementary Properties of the Maximum of Two Elements, and hence $|x_{k}|\le R_{t}$ for every $k$. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, claim 1 of Nonnegativity of Squares in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field and claim 3 of Properties of Finite Sums,

so $\lVert x\rVert\le R_{t}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By claims 2 and 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $d_{E}(0_{\mathbb{R}^{N}},x)=\lVert(-1)x\rVert=\lVert x\rVert\le R_{t}$, so $K_{t}$ is bounded in $(\mathbb{R}^{N},d_{E})$.

Moreover, for $x\in K_{t}$ and $(i,j)\in\Pi_{N}$, (2.1) with $\mu=\mu_{0}$ and $A(x)\le R_{t}$ give $T_{ij}(x)\ge\ell_{t}$, where $\ell_{t}=-2\beta\mu_{0}^{-1}R_{t}+\beta(1-\log\mu_{0})$. Fix $(i_{0},j_{0})\in\Pi_{N}$. Applying Nonnegativity and Monotonicity of a Sum over a Finite Index Set §monotone to the nonnegative function $p\mapsto T_{p}(x)-\ell_{t}$ with $E=\{(i_{0},j_{0})\}$ (and claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set), and then claims 3 and 4 of Properties of a Sum over a Finite Index Set,

Since $V_{1}(x_{k})\ge a_{0}|x_{k}|-b_{0}\ge-b_{0}$, we have $\sum_{k=1}^{N}V_{1}(x_{k})\ge-Nb_{0}$ and so $H(x)=P(x)-\sum_{k=1}^{N}V_{1}(x_{k})\le t+Nb_{0}$. Hence $-\beta\log(x_{i_{0}}-x_{j_{0}})\le L_{t}$ with $L_{t}=t+Nb_{0}-n_{\Pi}\ell_{t}+\ell_{t}$, that is, $\log(x_{i_{0}}-x_{j_{0}})\ge-\beta^{-1}L_{t}$. Put $\delta_{t}=\exp(-\beta^{-1}L_{t})$, which is positive by claim 2 of Basic Properties of the Exponential Function. As $\exp$ is strictly increasing (claim 4 there), $\exp(u)\ge\exp(w)$ whenever $u\ge w$, so by The Natural Logarithm, $x_{i_{0}}-x_{j_{0}}=\exp(\log(x_{i_{0}}-x_{j_{0}}))\ge\delta_{t}$. Thus

(2d) Let $(x^{m})_{m\in\mathbb{N}}$ be a sequence in $K_{t}$ converging to some $x\in\mathbb{R}^{N}$ in $(\mathbb{R}^{N},d_{E})$. For $k\in[N]$, the $k$th coordinate of $x^{m}-x$ is $x^{m}_{k}-x_{k}$ (Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$), so $|x^{m}_{k}-x_{k}|\le\lVert x^{m}-x\rVert=d_{E}(x^{m},x)$ by claims 4 and 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$; the real sequence $(d_{E}(x^{m},x))_{m}$ converges to $0$ by the definition of convergence, so $(x^{m}_{k})_{m}$ converges to $x_{k}$ by claim 3 of Order Properties of Limits of Real Sequences. For $(i,j)\in\Pi_{N}$, $(x^{m}_{i}-x^{m}_{j})_{m}$ converges to $x_{i}-x_{j}$ by claim 3 of Arithmetic of Limits of Real Sequences, and (2.4), Constant Sequences and Index-Shifted Sequences of Real Numbers §constant and claim 1 of Order Properties of Limits of Real Sequences give $0<\delta_{t}\le x_{i}-x_{j}$, so $x_{j}<x_{i}$. Hence $x\in W_{N}$ by The Weyl Chamber of Ordered Points in Euclidean Space. By claim 1 and Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §value (with $k=2$), $P$ is continuous at $x$ relative to $W_{N}$; the sequence $(x^{m})$ lies in $W_{N}$ and converges to $x$ for the restricted metric, which has the same distances, so by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential the sequence $(P(x^{m}))_{m}$ converges to $P(x)$ in $(\mathbb{R},d_{\mathbb{R}})$, which is convergence of real sequences since $d_{\mathbb{R}}(s,s')=|s-s'|$ (The Real Line: Standing Notation and Background for Calculus §metric). As $P(x^{m})\le t$ for every $m$, Constant Sequences and Index-Shifted Sequences of Real Numbers §constant and claim 1 of Order Properties of Limits of Real Sequences give $P(x)\le t$, so $x\in K_{t}$. By Sequential Characterization of Closed Subsets of a Metric Space, $K_{t}$ is closed in $\mathbb{R}^{N}$; being also bounded by (2c), it is compact by Heine-Borel Theorem in $\mathbb{R}^n$. Hence clause 2 of Penalty on an Open Subset of Euclidean Space holds and $P$ is a penalty on $W_{N}$.

Claim 3. Assume $0\le V_{1}''(t)$ for every $t\in\mathbb{R}$; by (0b), $V_{1}'$ is nondecreasing on $\mathbb{R}$. Let $x,y\in W_{N}$. By Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, Gradient of a Real-Valued Function on a Euclidean Open Set, claim 1, distributivity and claim 2 of Properties of Finite Sums,

The first term is nonnegative by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §monotone. For each $k$: if $y_{k}\le x_{k}$ then $V_{1}'(y_{k})\le V_{1}'(x_{k})$, so both factors of the $k$th summand are nonnegative (claim 3 of Elementary Arithmetic in an Ordered Field); if $x_{k}<y_{k}$ the summand equals $\bigl(V_{1}'(y_{k})-V_{1}'(x_{k})\bigr)(y_{k}-x_{k})$, again a product of nonnegative factors. A product of nonnegative numbers is nonnegative by claim 5 of Elementary Arithmetic in an Ordered Field, so the sum is nonnegative by claim 5 of Properties of Finite Sums, and the whole expression is nonnegative by claim 2 of Elementary Arithmetic in an Ordered Field.

Claim 4. Assume the hypotheses. By claim 2, $P$ is a penalty on $W_{N}$, so by Basic Properties of the Sublevel Sets of a Penalty §bounded-below there is $m\in\mathbb{R}$ with $m\le P(x)$ for every $x\in W_{N}$. Put

where $\eta$ is positive since $M\ge2>0$ (claim 7 of Elementary Order Arithmetic in an Ordered Field); let $C_{\eta}$ be given by the regular growth hypothesis for this $\eta$, let $C^{+}=\max\{C_{\eta},0\}$, which satisfies $0\le C^{+}$ and $C_{\eta}\le C^{+}$ by claim 1 of Elementary Properties of the Maximum of Two Elements, and put $C=(\tfrac{\kappa}{2}N+\beta N^{2})C^{+}+\lambda|m|$. These constants do not depend on $x$.

(4.0) From $0\le\kappa<2\beta$ we get $0<2\beta-\kappa\le2\beta$, hence $0<\varepsilon\le\tfrac12\le1$, $1-\varepsilon=(2\beta+\kappa)(4\beta)^{-1}\ge\tfrac12$, and

Since $8\beta N\le M$ and $2\kappa\le M$ (all summands of $M$ being nonnegative), $8\beta N\eta\le M\eta=1$ and $2\kappa\eta\le1$, so $2\beta N\eta\le\tfrac14$ and $\tfrac{\kappa}{2}\eta\le\tfrac14$. Hence $(1-\varepsilon)(1-2\beta N\eta)\ge\tfrac12\cdot\tfrac34=\tfrac38$ and

Now fix $x\in W_{N}$ and abbreviate $v_{k}=v_{k}(x)$, $a_{kj}=a_{kj}(x)$, $S=S(x)$; by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\lVert v(x)\rVert^{2}=\sum_{k=1}^{N}v_{k}^{2}$.

(4a) We show that for all $k,j\in[N]$,

If $k=j$ the left side is $0$ and the right side is nonnegative (claim 2 of Nonnegativity of Squares in an Ordered Field). If $k\ne j$ then $x_{k}\ne x_{j}$; let $a$ be the smaller and $b$ the larger of $x_{j},x_{k}$, so $a<b$. Then $a_{kj}(v_{k}-v_{j})=\bigl(V_{1}'(b)-V_{1}'(a)\bigr)(b-a)^{-1}$: for $a=x_{j}$ this is the definition of $a_{kj}$, and for $a=x_{k}$ both factors change sign. By (0a) there is $\xi$ with $a<\xi<b$ and $a_{kj}(v_{k}-v_{j})=V_{1}''(\xi)$. Since the hypothesis of claim 3 holds, (0b) gives $p\le u\le q$ with $p=V_{1}'(a)$, $u=V_{1}'(\xi)$, $q=V_{1}'(b)$, and $\{p,q\}=\{v_{j},v_{k}\}$. If $0\le u$ then $0\le u\le q$ and $u^{2}\le q^{2}$; if $u<0$ then $0\le-u\le-p$ (claim 4 of Elementary Order Arithmetic in an Ordered Field) and $u^{2}=(-u)^{2}\le p^{2}$; both by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. As $p^{2},q^{2}\ge0$, in either case $u^{2}\le v_{k}^{2}+v_{j}^{2}$, and regular growth gives

(4b) Summing (4.1) over $j$ and then over $k$ termwise, with $\sum_{j=1}^{N}(C^{+}+\eta v_{k}^{2}+\eta v_{j}^{2})=NC^{+}+N\eta v_{k}^{2}+\eta\lVert v(x)\rVert^{2}$,

(4c) By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, the last paragraph of the proof of claim 1, Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$ and claims 2 and 3 of Properties of Finite Sums,

By The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §calogero, $\lVert DH(x)\rVert^{2}=\beta^{2}S$, and by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §symmetrisation with $z=v(x)$ and claim 8 of Elementary Order Arithmetic in an Ordered Field, $2\,DH(x)\cdot v(x)=-\beta Q$. Multiplying (4b) by $-\beta<0$,

(4d) By claim 1, and summing the regular growth bounds $V_{1}''(x_{k})\le C_{\eta}+\eta v_{k}^{2}\le C^{+}+\eta v_{k}^{2}$ over $k$,

(4f) Multiplying (4.3) by $\tfrac{\kappa}{2}\ge0$ and (4.2) by $1-\varepsilon>0$ (claim 5 of Elementary Arithmetic in an Ordered Field) and subtracting,

Here $S\ge0$ and $\lVert v(x)\rVert^{2}\ge0$ (claim 2 of Nonnegativity of Squares in an Ordered Field, claim 5 of Properties of Finite Sums), and both brackets are negative by (4.0), so the first two terms are nonpositive; and $(1-\varepsilon)\beta N^{2}C^{+}\le\beta N^{2}C^{+}$ since the difference $\varepsilon\beta N^{2}C^{+}$ is nonnegative. Therefore

(4g) Finally $-|m|\le m\le P(x)$ by claim 3 of Properties of the Absolute Value in an Ordered Field, so $0\le\lambda P(x)+\lambda|m|$ since $0<\lambda$ (claim 5 of Elementary Arithmetic in an Ordered Field). Adding this to the last display gives

with $0<\varepsilon\le1$ and $C$ independent of $x$, as claimed.