# Product Entropy and Recycling

**Information entropy** came up in an earlier post about the **maximum entropy method** applied to **species modeling**. There I described entropy (**H**) as a measure of how spread out a probability distribution is. It can also be viewed as the information content of a message, as in how many bits are required to encode the message. As a simple example, consider a restaurant menu having four equally likely options with the customer’s order being the message. Then H = 2. **Claude Shannon** developed much of the theory around this type of entropy, sometimes called the Shannon Information (see his 1948 paper “The Mathematical Theory of Communication”).

Gutowski and Dahmus connect information entropy to product recycling in their 2005 ISEE paper “Mixing Entropy and Product Recycling”. Entropy is used to measure the “complexity” of the mixture of materials in a given product being recycled. It provides a lower bound for the average number of separation steps required to isolate the materials in the product. Perhaps a stronger case could have been made for why the entropy is a more useful and practical measure than just the number of materials in the product.

The paper contains an interesting plot (reproduced below) of product material value version material dispersion (i.e. entropy) for a number of products. The authors point out that

… design trends constantly move products towards the lower right hand corner of the figure.

So that is towards low material value, highly mixed (i.e. high entropy) products. See the toaster posts for a good example of this.

They continue:

Designers are motivated to constantly replace current materials with less expensive materials and less of them, and to increase functionality, often by using a greater variety of materials in a product.

They go on to state that large resources are invested in product development that effectively leads to towards lower recyclablility, while only modest ones go towards pushing products in the other direction.

Note also that the plot illustrates the natural divide between recycled and non-recycled products. It also suggests which currently non-recycled products, such as televisions, are closest to the boundary. These are the products that show the most promise for crossing the “recycling boundary”, and this could be achieved by lowering the complexity of their material mixtures. Again, this goes back to **cradle-to-cradle** type ideas of designing products to be more readily and safely broken into their components, which can be then reused in new products or left to biodegrade.

more on entropy to come …