The motion of particles at thermal equilibrium usually has a finite correlation time and will eventually be randomized after a long delay time,so that their displacement follows the Gaussian statistics,as dictated by the central limit theorem.Non-Gaussian fluctuations with an exponential tail in their probability density function(PDF),on the other hand,are often observed in non-equilibrium steady states(NESSs)and one does not understand why they appear so often.In this talk,I will present two examples of NESSs,in which we find the exponential PDF is directly linked to dynamic heterogeneity in the system.One example is turbulent Rayleigh-Bénard convection(RBC),in which warm fluid rises,cold fluid falls,and their mixing produces convective turbulence.The measured PDF P(δT)of temperature fluctuations δT in the central region of the flow was found to have a long exponential tail [1].The other example is the lateral motion of acetylcholine receptors(AChRs)on live muscle cell membrane.The AChRs are linked to fluorescent quantum dots and their trajectories are tracked with nanometer accuracy over a broad range of times.The measured PDF of the protein displacements showed a consistent exponential tail,which is robust and universal for cells under different conditions [2].In both cases,we found that because of the dynamic heterogeneity,the unconditional PDF P(x)can be written as a weighted convolution of a set of fluctuation modes G(x|ε)conditioned on a constant dynamic variable ε.For RBC,the correct dynamic variable ε is the thermal dissipation rate.For protein diffusion,it is the molecular diffusivity.The conditional PDF G(x|ε)under a constant ε is found to be of Gaussian form and its variance σ2(ε)for different values of ε follows an exponential distribution.The convolution of the two distribution functions gives rise to the exponential PDF P(x).Our work thus provides a physical mechanism of the observed exponential PDF in RBC and protein diffusion.It also sheds light on the origin of non-Gaussian fluctuations in other NESSs.