877 research outputs found
On an Extension Problem for Density Matrices
We investigate the problem of the existence of a density matrix rho on the
product of three Hilbert spaces with given marginals on the pair (1,2) and the
pair (2,3). While we do not solve this problem completely we offer partial
results in the form of some necessary and some sufficient conditions on the two
marginals. The quantum case differs markedly from the classical (commutative)
case, where the obvious necessary compatibility condition suffices, namely,
trace_1 (rho_{12}) = \trace_3 (rho_{23}).Comment: 12 pages late
Inequalities that sharpen the triangle inequality for sums of functions in
We study inequalities that sharpen the triangle inequality for sums of
functions in
Inequalities for Lᵖ-Norms that Sharpen the Triangle Inequality and Complement Hanner’s Inequality
In 2006 Carbery raised a question about an improvement on the naïve norm inequality ∥f+g∥^p_p ≤ 2^(p−1)(∥f∥^p_p+∥g∥^p_p) for two functions f and g in Lᵖ of any measure space. When f=g this is an equality, but when the supports of f and g are disjoint the factor 2^(p−1) is not needed. Carbery’s question concerns a proposed interpolation between the two situations for p > 2 with the interpolation parameter measuring the overlap being ∥fg∥_(p/2). Carbery proved that his proposed inequality holds in a special case. Here, we prove the inequality for all functions and, in fact, we prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all real p ≠ 0
On the (Boltzmann) Entropy of Nonequilibrium Systems
Boltzmann defined the entropy of a macroscopic system in a macrostate as
the of the volume of phase space (number of microstates) corresponding
to . This agrees with the thermodynamic entropy of Clausius when
specifies the locally conserved quantities of a system in local thermal
equilibrium (LTE). Here we discuss Boltzmann's entropy, involving an
appropriate choice of macro-variables, for systems not in LTE. We generalize
the formulas of Boltzmann for dilute gases and of Resibois for hard sphere
fluids and show that for macro-variables satisfying any deterministic
autonomous evolution equation arising from the microscopic dynamics the
corresponding Boltzmann entropy must satisfy an -theorem.Comment: 31 pages, in Tex, authors' e-mails: [email protected],
[email protected]
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