Physics 301 3-Dec-2004 30-1Transp ortWhen a system is not in equilibrium, various transport processes may occur. By a“transport process” is meant the transfer of energy, charge, particles, etc. by the physi-cal motion of parti cles. In some cases, wave phenomena may be important in transportprocesses such as heat conduction in an electrical insulator. The fact that the materialis an insulator means that the electrons are not free to move. The conduction of heatoccurs by atoms passing energy from one to the next. So there is no net flow of atoms,but there is a flow of heat energy. This is similar to what happens with sound waves. Ofcourse, sound waves can be thought of as phonons. . .. In the case of electric current, chargeis transp orted by the physical movement of particles. In an electrical conductor, heat istransported by the “wavelike” action described earlier and also by the flow of electrons.We will consider systems which are only mildly out of equilibrium. Usually t hesesystems will be in a steady state in which nothing varies with time. Of course, this is anidealization—if we consider heat conduction by a system connecting two thermal reservoirs,then the reservoirs are a ssumed to be so large that the removal of heat energy from onereservoir and its addition to the other makes negligible change in the reservoirs.If the system is only slightly out of equilibrium, it usually happens that a flow will beset up in such a direction that it would restore equilibrium. For example, consider a rodconnecting two thermal reservoirs at different temperatures. Heat energy flows t hroughthe rod from the high temperature reservoir to the low t emperature reservoir. This isthe direction required to move closer to equilibrium. Note that since we’re discussinga non-equilibrium situation, the system tries to maximize or minimize the appropriatethermodynamic function (to get to equil ibrium). In this case, we have reservoirs in thermalcontact (through the rod). No work can be done and no particles can be exchanged. So theappropriate thing to do is to try and ma ximize the entropy. Sure enough, energy leavinga high temperature reservoir removes some entropy from that reservoir, but when thatsame energy is deposited in the low temperature reservoir, more entropy is added thanwas removed from the high temperature reservoir. So the system is creating entropy whichis what you do if you want to maximize i t! In other situations, the flow might be such thatit would minimize the free energy.Since we are speaking of a flow in response to a non-equilibrium situation, it seemsreasonable that the “strength” of the flow should be proportional to the departure fromequilibrium, at least for small departures from equilibrium. In other words, we’re go ingto assume that the flow can be expanded in a Taylor series in whatever measures thedeparture from equilibrium. The constant term must be zero (no flow at equilibrium) sothe first term must be the linear term. We’ll assume that the linear term is non-zero anddepartures from equilibrium are small enough t hat we only need consider the linear term.How do we measure a flow? An electric current density is an example. Recall thatthe electric current density is a vector which has the dimensions o f charge per unit a reaCopyrightc 2004, Princeton Uni versity Physics Department, Edward J. Gro thPhysics 301 3-Dec-2004 30-2per unit time. In general, we’ll consider the flux density which is a vector with units ofwhatever is being transport ed per unit area per unit time. The direction of the vector givesthe direction in which the whatever is being transported. The magnitude gives the netamount of whatever that crosses a unit area perpendicular to the direction of the vectorin a unit time. In the case of electric current density, negative charges moving with somevelocity in some direction produce the same current as positi ve charges moving with thesame velocity in the opposite direction. The current or flux density is al ways the netcurrent or flux density. Sometimes we want the current or flux crossing a surface. Thishas units of what ever per unit time. It is found by integrating the current density or fluxdensity times the unit normal over the surface,I =ZsurfaceJ · n dA ,where I is the current or flux, J is the current density or flux density, n is the unit normalto the surface pointing from the negative side to the positive side of the surface, and dAis the differential area element on the surface.Food for thought: suppose the quantity being transported is i tself a vector. Howwould you describe the flux density?How do we measure a departure from equili brium? In general, any non-uniformitiesin a system indicate non-equilibrium. For example, temperature variat ions, variations inparticle concentrations, or variations in electric potential might indicate a non-equilibriumsituation. But there’s more to it than just a varia tion; there is also the scale of thevariation. A 10, 000 Vol t potential difference between two electrodes in air is not a bigdeal if the electrodes are separated by several meters. Rather dramatic effects occur ifthey’re separated by only a millimeter—probably more than the linear term is needed todescribe the resulting transport! In other words, how rapidly the non-uniformity varieswith position is the important quantity for determining the transport.The upshot of all this hand waving is that a transport process is described byJ = (Constant) × [∇(Scalar Field)] .Some transport laws are the following: Ohm’s law relating electric current density, Jqto the electric field or the gra dient of the electric potential where the coefficient is theconductivity, σ (not the entropy nor the cross section, here),Jq= σE = −σ∇Φ .The negative sign indicates that (positive) charge flows from high to low potential. For par-ticle diffusion there is Fick’s law relating the particle flux density, Jn, to the concentration.The proportionality constant i s called the diffusivity, D,Jn= −D∇n .Copyrightc 2004, Princeton Uni versity Physics Department, Edward J. Gro thPhysics 301 3-Dec-2004 30-3Again, the negative sign indicates that particles diffuse from high to low concentrations.Note that diffusivity has dimensions of area per unit time or length times velocity. Fo urier’slaw describes heat conduction. It relates the flow of energy (heat), Juto the temperaturegradient. The proportio nal ity constant is the