Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations

Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.     

Synthesis and Structure Characterization of 2‐(Dimethylaminomethyl)pyrrolate and 2,5‐Bis(dimethylaminomethyl)pyrrolate Zirconium Complexes

Tetrakis(diethylamido)zirconium reacts with 2‐(dimethylamino)methyl pyrrole (DMAMP) and 2,5‐[bis(dimethylamino)methyl]pyrrole (BDMAMP) to give Zr(NEt)₂(DMAMP)₂ 1 and Zr(NEt)₃(BDMAMP) 2 , respectively. Both 1 and 2 have been characterized by ¹H and ¹³C NMR spectroscopies and 1 has also been characterized by X‐ray crystallography. Complex 1 shows an agostic interaction between Zr and H(21A) in solid state that is not sustained in solution. Reacting 1 with 2 equivalents of trimethylsilyl chloride in toluene yields ZrCl₂(DMAMP)₂ 3 in 75% yield which was characterized by ¹H and ¹³C NMR spectroscopies.     

The Nitrogen‐Assisted Triphenylphosphine Displacement of 3‐Hydroxypyridine and 3‐Aminopyridine Ligands in Palladium(II) Complexes: Crystal Structures of [Pd(PPh3)Br]2{μ,η2‐C5H3N(OH)}2 and [Pd(PPh3)Br]2{μ,η2‐C5H3N(NH2)}2

Treatment of Pd(PPh₃₎₄ with 2‐bromo‐3‐hydroxypyridine [C₅H₃N(OH)Br] and 3‐amino‐2‐bromopyridine [C₅H₃N(NH₂)Br] in dichloromethane at ambient temperature cause the oxidative addition reaction to produce the palladium complex [Pd(PPh₃₎₂{η¹‐C₅H₃N(OH)}(Br)], 2 and [Pd(PPh₃₎₂{η¹‐C₅H₃N(NH₂)}(Br)], 3 , by substituting two triphenylphosphine ligands, respectively. In dichloromethane solution of complexes 2 and 3 at ambient temperature for 3 days, it undergo displacement of the triphenylphosphine ligand to form the dipalladium complexes [Pd(PPh₃)Br]₂{μ,η²‐C₅H₃N(OH)}₂, 4 and [Pd(PPh₃)Br]₂{μ,η²‐C₅H₃N(NH₂)}₂, 5 , in which the two 3‐hydroxypyridine and 3‐aminopyridine ligands coordinated through carbon to one metal center and bridging the other metal through nitrogen atom, respectively. Complexes 4 and 5 are characterized by X‐ray diffraction analyses.     

Synthesis,Crystal Structure,and Magnetic Properties of (2,2′‐dipyridylamine)(X−) Copper(II) Complexes (X−: Cyanate,Dicyanamide, Acetate,Thiocyanato)

The synthesis, structure, and magnetic properties of four 2,2′‐dipyridylamine ligand (abbreviated as Hdpa) containing copper(II) complexes. There is one binuclear compound, which is [Cu₂(μ_1,1‐NCO)₂(NCO)₂(Hdpa)₂] ( 1 ), and three mononuclear compounds, which are [Cu{N(CN)₂}₂(Hdpa)₂] ( 2 ), [Cu(CH₃CO₂)(Hdpa)₂·N(CN)₂] ( 3 ), and [Cu(NCS)(Acac)] ( 4 ). Compounds 1 and 4 crystallize in the monoclinic system, space group P2(1)/c and Z = 4, with a = 8.2465(6) Å, b = 9.3059(7) Å, c = 16.0817(12) Å, β = 91.090(1)°, and V = 1233.90(16) ų for 1 and a = 7.6766(6) Å, b = 21.888(3) Å, c = 10.4678(12) Å, β = 90.301(2)°, and V= 1758.8(4) ų for 4 . Compounds 2 and 3 crystallize in the triclinic system, space group P‐1 and Z = 1, with a = 8.1140(3) Å, b = 8.2470(3) Å, c = 9.3120(4) Å, β = 102.2370(10)°, and V = 592.63(4) ų for 2 and a = 7.4780(2) Å, b = 12.5700(3) Å, c = 13.0450(3) Å, β = 96.351(2)°, and V = 1211.17(5) ų for 3 . Complex ( 1 ), the magnetic data was fitted by the Bleaney‐Bowers equation (1). A very good fit was derived with J = 23.96, Θ = ?1.5 (g = 1.97). Complex ( 1 ) shows the ferromagnetism. Complexes ( 2 ), ( 3 ) and ( 4 ) of have the it is the typical paramagnetic behavior of unpaired electrons. Under a low temperature around 25 K, complexes ( 2 ) and ( 3 ) show weak ferromagnetic behavior. They are the cause of hydrogen bonds.