The electronic density is defined with respect the density in the leads: n$_{e}$ ( x, t ) − n$_{e}$ ≡ 1 √ $_{π}$∂$_{x}$ ϑ$_{e}$ ( x, t ) + n$_{e}$ [1 − h(x,d) 2 ] − n$_{e}$ = 1 √ $_{π}$∂$_{x}$ ϑ$_{e}$ ( x, t ) − n$_{e}$ h(x,d) 2 . The electrostatic potential δv ( x, t ) is determined by the Poisson equation [34]: κ is the dielectric constant and ξ is the gate screening length [27]. Next we will compute the two components currents ∂$_{x}$ϕ ( n =0) e ( x ) and ∂$_{x}$ϕ ( n =0) e ( x ). The non-zero mode part ∂$_{x}$ϕ ( n =0) e ( x ) is induced by the non-uniform ground state − n$_{e}$ h(x,d) 2 . This current will be obtained within the linear response theory [36], where we have replaced the regular commutator by the Dirac commutator : << F | ... | F >>$_{T}$ stands for thermodynamic expectation value. Using the low energy Bosonic spectrum, which emerges from equation (27), we approximate for times t obeying v$_{eff.}$ 2 π L t < 1, the time integration by a constant time τ if v$_{eff.}$ 2 π L τ < 1. At finite temperature, τ is given by τ = smallest [ L $_{v$_{eff.}$}$, L$_{T}$ $_{v$_{eff.}$}$] ≡ L$_{T,L}$ v$_{eff.}$ The zero mode current is computed using the current operator ∂$_{x}$ϕ ( n =0) e ( x ). At finite temperatures static electric current will be given by the thermodynamic expectation value of the operator N$_{↑}$$_{,L}$ + N$_{↓}$$_{,L}$ − N$_{↑}$$_{,R}$ − N$_{↓}$$_{,R}$ L . This expectation value is determined by the thermal reservoirs. The thermal function is represented by << F | ( N$_{σ,L}$ − N$_{σ,R}$ ) | F >>$_{T}$ ( T stands for the thermal occupation values). For the wire region, we have the Hamiltonian H ( n =0) wire = hv 2 L ∑ σ = ↑ , ↓ ( d $_{L}$)[ N 2 σ,L + N 2 $_{σ,R}$] which is subjected to the constraint of exclusion of double occupancy. For the leads, we have the unconstrained Hamiltonian H ( n =0) leads = hv 2 L ∑ σ = ↑ , ↓ ( L − d L )[ N 2 σ,L + N 2 $_{σ,R}$]. In the limit of long leads d L << 1, the thermal weight of the region d is negligible in comparison to the leads region. Therefore, the thermal expectation value is given by the unconstrained Hamiltonian: We compute the partition function under the assumption that d L << 1. Therefore, the constraint partition function Z = T$_{r}$ [ $_{e}$− H ( n =0) wire + H ( n =0) leads + H n =0) reservoir K B T ] can be replaced by the un- 15 H ( n =0) ≈ hv 2 L ∑ σ = ↑ , ↓$^{[}$$^{N}$$^{2}$ σ,L + N 2 $_{σ,R}$] + ∑ σ = ↑ , ↓$^{[}$$^{µ$_{L}$}$$^{N$_{σ,L}$}$ + µ$_{R}$N$_{σ,R}$ ] (36) δ << F | ∂$_{x}$ ˆ ϕ$_{e}$ ( x, t )) | F >>$_{T}$ ≡ << F | ∂$_{x}$ ˆ ϕ$_{e}$ ( x, t ) | F >>$_{T,δv}$$_{(}$$_{x,t}$$_{)}$ − << F | ∂$_{x}$ ˆ ϕ$_{e}$ ( x, t ) | F >>$_{T}$ = i ¯ h ∫ L 2 − L 2 dy ∫ t 0 dt ′ << F | [( − e ) δv ( y, t ′$_{)(}$ 1 √ π ∂$_{y}$ϑ$_{e}$ ( y, t ′$_{)}$ − n$_{e}$ h(y,d) 2 ) , ∂$_{x}$ ˆ ϕ$_{e}$ ( x, t )]$_{Dirac}$ | F >>$_{T}$ ≈ τ en$_{e}$ ¯ hκ [1 − h(x,d) 2 ] ∫ L 2 − L 2 dy h(y,d) 2 ∂$_{x}$ [ 1 √ ( x − y ) 2 + a 2 − 1 √ ( x − y ) 2 + ξ $_{2}$] (35) δv ( x, t ) = − e κ ∫ L 2 − L 2 dy [ 1 √ ( x − y ) 2 + a 2 − 1 √ ( x − y ) 2 + ξ $_{2}$][ 1 √ π ∂$_{y}$ϑ$_{e}$ ( y, t ) − n$_{e}$ h(y,d) 2 ] (34)